Theory and design of recoil systems and gun carriages

UNIVERSITY OF CALIFORNIA

GIFT OF

Col. Glen F.

PE.PRRTr«QE.1MT DOCUMENT NO. ZO35

THEORY AND DESIGN

OF

RECOIL SYSTEMS

AND

GUN CARRIAGES

Prepared in the
Office of the Chief of Ordnance

SEPTEMBER,

REPRODUCTION PUANT
WA5H\NQTON BARRRCKS O.C.

-rinf
Lilrary

U.F

ORDNANCE DEPARTMENT
Document No. 2035

Off ic« of the Chief of Ordnance

WAR DEPARTMENT,

Washington, October 1921.

The following publication entitled "Theory and
Design of Recoil Systems and Gun Carriages" is pub-
lished for the information and guidance of all students
of the Ordnance training schools, and other similar
educational organizations. The contents should not be
republished without Authority.

By order of the Secretary of

C. C. WILLIAMS,

MAJOR GENERAL, CHIEF OF ORDNANCE.

Jjlraiy

FOREWORD

This edition is published in its present form with
lioeral margins and spacing so that corrections or
additions may be freely made. In a document of this
kind it is almost inevitable that ambiguities, errors
and misstatenents will appear, and it is only in extended
and repeated use that these are fully exposed. It will
therefore be appreciated if all those to whom this
volume comes and who use it critically will forward
criticisms, corrections or necessary additions to the
Artillery Division, Ordnance Office, Washington, D. C.,
so that these nay be incorporated in the master volume.
After such changes have been received for a suitable
period, it is expected to have the text printed in
usual book form.

PREFACE

Although strictly artillery design may
be considered a highly specialized branch of
machine design, there are so many features that
differentiate this work from ordinary machine
design, it has been felt that a volume covering
the specialized points is of fundamental
importance in order that our designing engineers
may have in a readily accessible from reference
data covering the general subject and in
particular those features of modern development
not now covered in published works. Such is the
purpose of this volume.

Artillery design may be subdivided into the
design of cannon, and the design of gun carriages
and recoil systems. During the late war the ex-
tensive introduction of self propelled gun mounts,
such as caterpillar vehicles, has introduced
automotive problems in the design of these types
of gun mounts in addition to the ordinary
consideration affecting design of gun mounts.
Further in the design of artillery we have three
important aspects, - (l) the technical and
theoretical considerations of a design, (2) the
fabrication, standardization and production
features, and (3) the service and field require-
ments to be fulfilled. All three aspects are
equally important and a successful design results
only from a balanced consideration of the three.

This discussion has been written under the
auspices of Colonel G. F. Jenks, Chief of the
Artillery Division, Ordnance Department, U. S. A.
and of Colonel J. B. Rose, Chief of the Mobile
Gun Carriage Section of that Division. Effort
has been made to arrange systematically in a
form for reference tne great quantity of
engineering data in the files of the office.
In order to develop and analyze this data, it has

been necessary to introduce a considerable number

of original discussions and deductions.

The work is an attempt to cover only the

*»
technical aspect of the design of gun carriages

and recoil systems. The fabrication and field
service phases, though of course inherently
coordinate in. a design are subjects of such com-
plexity and broadness that they require for their
full appreciation a separate treatment. These
aspects have therefore necessarily been entirely
onitted, except in so far as they are directly
connected with the technical features involved.
Acknowledgement and thanks are especially
due to Colonel J. B. Rose, who has proof read the
complete work in the view of bringing the data into
conformity with the practice and standards of the
Ordnance Department. It should be stated, however,
that this has been done only to the degree which
was found possible without destroying original con-
clusions and discussions or without alteration of
the system of nomenclature used. The latter is in
partial but not complete agreement with the most
general practice. Further acknowledgement and
thanks for suggestions on the various parts of the
work are due to: -

Mr. D. A. Gurney, Ordnance Engineer, Mobile
Gun Carriage Section,
Artillery Division.
Prof. B. V. Huntington, Professor of

Mathematics and Mechanics,
Harvard University

Professor C. E. Fuller, Professor of Ap-
plied Theoretical
Mechanics, Massachusetts
Institute of Technology.
Professor G. Lanza, Professor Emeritus

in charge of Mechanical
Engineering Department,
Massachusetts Institute
of Technology.

Acknowledgement of assistance on the Computation
work is due to Mr. E. V. B. Thomas, Mr. Kasargian
and Mr. McVey of the Artillery Division, also to
Messrs. Murray H. Resni Coff and 0. L. Garver for
preparing this data for publication.

RUPBN BKSERGIAN,
Formerly Captain, Ordnance Dept.U.S.A,

Chapter I

Chapter II

Chapter III
Chapter IV

Chapter V

Chapter VI

Chapter VII

Chapter VIII
Chapter IX

Introduction and Elements - Types of
Cannon and Carriages - Classifi-
cation of Carriages and Recoil
Systems.

Dynamics of Interior Ballistics as Af-
fecting Recoil Design - The Para-
bolic Trajectory.

External Reaction on Carriage during
Recoil - Stability - Jump.

Internal Reactions throughout a
Carriage during Recoil and
Counter Recoil.

Gun

Hydraulic Principles as Applied to
Various Systems of Recoil and
Counter Recoil - General

Theories on Orifices and
Flow of Oil.

The Dynamics of Recoil and Counter Re-
coil - Differential Equations of
Resistance, Braking, etc. and
of Velocity - General
"Formulas for Recoil

and Counter Recoil,

Classification of Recoil Systems -

Derivation of General Formulas for
Design and Computation -

General Linitations, etc.

Hydro-pneumatic Recoil Systems.

Hydro-pneumatic Recoil Systems
(Continued)

8

Chapter X Railway Gun Carriages.
Chapter XI Gun Lift Carriages.
Chapter XII Double Recoil Systems.

Chapter XIII Miscellaneous Problems —

Discussions of Various Types
of Carriages.

CHAPTER I

TYPES AND PRINCIPAL ELEMENTS OF CANNON AND
CARRIAGES.

The fundamental principles of gun
carriage design are entirely the same
as those of engine and machine design,
and it is the object of this volume
merely to bring out the specific ap-
plication of these principles to the design of gun
carriages .

A gun carriage is a machine exercising primarily
the following functions:

(1) To provide a fixed firing platform
which dissipates the energy given to
the recoiling parts in reaction to the
energy imparted to the projectile and
powder gases.

(2) To return the recoiling parts to their
initial position for further firing.

(3) To provide the mechanism for elevat-
ing the gun for different ranges and
angles of site, and for traversing the
gun for changes in the direction of fire.

The effect of allowing the gun and a part of the
carriage to recoil is to reduce many times the stresses
in the carriage and to maintain its equilibrium. A
properly designed recoil system will give reactions
consistent with the strength and stability of the
carriage, and a smoothness of action which is essent-
ial for long service and accuracy. The success of one
design over another is due to perfection of many de-
tails, which insures smooth action and long service
and to a judicious compromise between many opposing
conditions and requirements.

To approach the study of carriage design, it is
necessary to know the elements of interior ballistics
and the characteristics of guns for meeting different

9

10

ballistic conditions in so far as these affect the
form of carriage and determine the forces acting
upon it. These subjects will, therefore, be
briefly considered, but a complete discussion must
be obtained from works treating them specifically.

From one view point a cannon may
be considered as a tube of proper
thickness for strength, having a
chamber in the rear of somewhat larg-
er diameter which contains the powder
charge. The powder charge is inserted by opening a
breech block in the rear end of the cannon. This
breech block necessarily must withstand the maximum
powder pressure over its cross section and a power-
ful locking device is therefore needed. The details
of this mechanism are complicated, but need not be
considered in carriage design, except in special
cases where the breech mechanism is operated during
counter recoil. The design of the rifling grooves
and capacity of powder chamber will be considered
later.

The elements of a gun are shown in figure (1).
"A" is the powder chamber, "B" the rifled portion
of the bore, "C" the breech block, "D" the gun lug
for the attachment of piston rods, which restrain
the gun in recoil.

— *

— V-

il \ .

-L
fV

^

The caliber of a gun is the diameter of the bore
and is expressed usually in millimeters or inches.
Speaking very roughly, small guns range from 37 m/m

11

to 75 m/m, and are suitable for mounting on
aeroplanes or for use with infantry. Light field
guns range from 75 m/m to 105 m/m. Ordinary medium
artillery ranges from 105 m/m to 8 inches* Heavy
artillery ranges from 8 inches upward. The above
classification refers to mobile field materiel only.

PQWTT7FRS AND flUNS Carriages are designed for either

howitzers or guns. Howitzers are
for high angle fire, the striking
angle being generally above 25 de-
grees. They have a medium or low

muzzle velocity. A gun is designed for range and, there-
fore, has a high muzzle velocity.

The angle of elevation of howitzers is usually
between 20"* and 70°and the muzzle velocities from 400
to 1800 feet per second. The angle of elevation of a
gun is usually from minus 5 degrees to plus 45 degrees
with muzzle velocities ranging from 1700 to 3000 feet
per second. The angular velocity of the projectile is
also considerably higher for a gun than for a howitzer.
In modern practice the line of demarcation between guns,
howitzers and mortars has become somewhat less distinct,
and we may consider all of them as cannon which decrease
in power in the order named and generally for use at
elevations which increase ia the order named.

Against aircraft, firing is at elevations from
0 to 80 degrees, hut the muzzle velocity is high; hence,
the pieces used in such work are properly classified
as "guns".

The traversing limitations of a gun and howitzer
may be the same or different but do not enter in the
differentiation between a gun and howitzer.

The muzzle velocity of howitzers being lower than
that of guns, it is possible with the same total weight
of materiel to fire a much heavier projectile.

12

BECOIU BG_PABT-g The recoiling parts consist of
the gun together with the various

parts attached to it and recoiling
with it. We have two methods of
arrangement of recoiling parts:

(1) the piston rods with their pistons
attached to the gun lug and recoiling
with the gua.

(2) the pistons and their rods held
stationary.

So far as the recoil mechanism is concerned we are
only concerned with the relative motion between the
rods and pistons and their cylinders.

The greater part of our guns in the service
translate in recoil directly along the axis of the
bore, others as on certain Barbette mounts and
double recoil systems have a translation in addition
to that along the axis of the bore. Guns on Dis-
appearing carriages and special other types have
rotation in addition to translation.

In ordinary recoil systems the center of gravity
of the recoiling parts is usually located slightly below
the axis of the bore. This insures a positive jump
(muzzle up) during the powder pressure period. If
the center of gravity of the recoiling parts is great-
ly below the axis of the bore considerable stresses
are brought upon the elevating rack and pinion, due
to the fact that the powder pressure causes an ex-
cessive turning effect about the trunnions the
amount depending also upon the location of these.
For this reason when the cylinders recoil with the
gun, extra weight is very often introduced on the
top of the gun. This, of course, raises the center
of gravity of the recoiling parts nearer the axis
of the bore.

The recoiling parts are constrained to recoil
parallel with the axis of the bore by gun clips en-
gaging in guides in a fixed cradle or by the gun
itself sliding in a fixed cylindrical sleeve. Due
to the fact that the braking forces developed in the

13

cylinders are usually considerably below the axis
of the bore during recoil, considerable pinching
action takes place at the front and rear clip contact
with the guides. This causes somewhat greater friction
than would be obtained by mere sliding friction.

The clips attached to the recoiling parts, or
rather to the gun itself, which in turn engage in the
guides of the cradle, are usually either continuous or
three to four in number. In order to maintain a con-
stant friction throughout the recoil, clips should
be evenly spaced along the gun and the front clip
should engage in the guides before the rear clip leaves
the guides. When the gun recoils in a sleeve or cy-
linder which is a part of the cradle, it is conetiroes
possible to distribute the various pistons and cy-
linders symmetrically about the axis of the bore.
As we shall see, this decreases the friction during the
recoil and counter recoil.

Figure (2) shows the recoiling mass where the
pistons and their rods recoil with the gun. Below
in figure (3) is shown a recoiling mass consisting
of the cylinders grouped together in a single forging
in a so-called slide or sleigh, and rigidly attached
to the gun.

14

. 3

nt£_CRADLE_ The cradle serves as a constrain-
ing member for the sliding of tlie gun

to the rear in recoil and as a support
for elevating the gun. The cradle and
the recoiling parts together are known
as the tipping parts and turn about horizontal trun-
nions fixed to the cradle and resting in bearings in
the top carriage. To elevate the tipping parts, an
elevating arc bolted to the cradle engages in a pinion
fixed to the top carriage, or vice versa. The cradle
and therefore the tipping parts are supported in the
top carriage at two points:

(1) at the trunnions
and

(2) at the tooth contact of the
elevating arc and pinion.

See figure (4).

IS

ng. 4-

When the cylinders do not recoil they are in turn
an integral part of the cradle, and therefore, the
recuperator forgings and the cradle are one and the
same. A sleigh may or may not he interposed between the
gun and cradle. With guns, where the cylinders recoil
with gun, the cradle merely serves the purpose of a
constraining guide for the recoiling parts and rigid-
ly attached to it are the piston rods and their pis-
tons .

TJPPIflG_FABT§ The term"tipping parts" applies

to those parts of a carriage which move
in the process of elevating the gun.
In order to rapidly elevate the gun, it
is considered very important that the
tipping parts are nicely balanced about the trunnions.
Thus the center of gravity of the tipping parts must
be located at the trunnions. As the height of the
trunnions and axis of the bore are governed by
stability at horizontal elevation, clearance in
traveling and accessibility for leading, the length
of recoil at maximum elevation becomes limited. If
a minimum elevation of about 20 degrees is allowed for
a howitzer, we might raise the trunnions, thereby in-

16

crease the length of recoil, and thus maintain

stability. When, however, a gun must fire at high ele-
vation as in antiaircraft materiel, or when a carriage
serves the double purpose of supporting a gun or a
howitzer at high elevations, the maximum possible re-
coil at maximum elevation becomes greatly limited.
The recoil displacement at maximum elevation may
be increased most satisfactorily by placing the
trunnions to the rear and introducing a balancing
gear for balancing tbe tipping parts about the
trunnions.

The balancing gear usually consists of an os-
cillating spring or pneumatic cylinder, the trunnions
of which rest in bearings in the top carriage, the
end of the piston rod being attached to the cradle.
Since it is difficult to obtain perfect balance by
this method throughout the elevation, the maximum
unbalanced moment in the process of elevation should
be considered in the design of the elevating gear
mechanism. A method by which exact balance can be
maintained throughout the elevation is obtained by
use of a cam and chain connecting tbe cradle with
the spring or pneumatic cylinder. In this case the
cam is fixed to the cradle and the spring cylinder to
the top carriage. However, due to the variation in
trunnion friction and other similar factors the
former method is probably better since a very close
approximation in balance throughout the elevation can
be obtained.

The reaction on the elevating arc and the
trunnion reaction are modified by the introduction
of tbe balancing gear, though ordinarily where the
weight of tipping parts is relatively small as com-
pared with the recoil reaction the effect of the
balancing gear on the reactions may be neglected.

When it is desired to use an independent line
of sight, a rocker is introduced between the
elevating pinion and cradle. The rocker, when moving,
is a part of the tipping parts. In the process of

17

elevating the gun an elevating pinion rotates

the rocker about the trunnions until the proper
line of sight is obtained; the cradle is then
brought into its proper position by gearing con-
necting the rocker and cradle.

TOP CARRIAGE The top carriage serves as an.
intermediary piece connecting the
tipping parts with the bottom car-
riage, or in semi-fixed mounts, with
the bottom platform. The top car-
riage is supported at its bottom by a vertical
pintle block and circular traversing clips. At
the top it supports the tipping parts on its trun-
nion bearings and elevating pinion bearing. The
top carriage -together with the tipping parts are
known as the traversing parts. To traverse the gun,
the top carriage with the tipping parts are rotated
in a horizontal plane about the pintle block by a
circular traversing reck and pinion or worm gear.
In certain types of field artillery the top
carriage is an integral part of the trail, in which
case traversing is obtained with respect to the
wheels and axle by moving the trail along the
axle and about the spade point as a pivot. Traverse
by this method is naturally very limited as com-
pared to traverse with a rotating top carriage. All
stationary mounts or field platform mounts have a
separate top carriage which serves this specific function
of traversing about the vertical pintle support. In very
large carriages the top carriage is supported by a
circular ring of horizontal rollers, the pintle
bearing merely serving as a constraining pivot. In
certain types where the bottom carriage itself is
traversed, the top carriage is used for translation
only. It is then supported on rollers moving
along an inclined or horizontal plane and the braking
is affected by a recoil cylinder in the top carriage which

18

connects the top carriage with the bottom carriage
through the piston rods.

Fig. 5

Top carriages way be roughly classified into:

(1) the ordinary type of side frames
connected at the front or rear by
cross beams or transoms which contain
the pivot bearing.

(2) pivot yoke type used on small
mobile mounts

and

(3) trail carriages.

The ordinary side frame type of top carriage is
extensively used on the stationary mounts on mobile
platform mounts and even on trail supported carriages.
The pivot yoke type is especially useful when split trails
are introduced, since it supports the equalizer bar for
balancing the distribution of the load between the
two trails.

19

TRAIL AND SPADE With mobile field artillery

it is customary to use a trail and
spade for the double purpose of
preventing a backward motion of the
carriage on firing of tbe gun, and
of giving sufficient stability to the carriage in
order that the wheels may not leave the ground.
We have two classifications of trails, - (1) the
single or box trail, (2) the split trail. With a
single trail it is necessary to have a large U-
shaped aperture or fork arrangement at the forward
end in order to elevate, load and traverse the gun
without interference. When split trails are used
we have two separate single trails which may turn
at the wheel ends about the axle. It is customary
with the split trails to introduce an equalizing

mechanism which connects the two trails and
distributes the load between the trails on
firing.

The spade and float support tbe trail and are
designed to take up the horizontal and vertical re-
actions at the rear end. In tbe design of the
spade and floats it is important that tbe unit bear-
ing pressure be held to a low value. This should
not be more than about 30 Ib. per sq. in. for the
float and 40 Ib. per sq. in. for the spade.

For wide traverse of the gun it is necessary
to lift the spade from the ground and turn the
carriage to the desired line of fire. For this
reason the static load on the spades should not ex-
ceed about 100 Ibs. for light carriages. Thus in
a preliminary lay-out of tbe carriage, it is
necessary to locate the center of gravity of tbe
total system in battery very close to the axle in
order that the static load under the float does not
exceed the desired amount. This inherently makes
the counter recoil stability in battery very small
especially at horizontal recoil and requires con-
siderable care in the design of a counter recoil

20

system.

At horizontal elevation the carriage is usually
designed with a very small margin of stability.
Therefore, in firing the vertical load on the float
practically equals the weight of the total system.
The bending moment in the trail gradually increases
from the spade toward the wheel axle. We have max-
imum bending moment at the attachment of the trail
to the top carriage or wheel axle.

PLATFORM MOUNTS. With fixed mounts and heavier
types of field artillery it is
customary to support the travers-
ing parts on a platform, that is,
the top carriage rests upon a

platform which serves as a bottom carriage. When
a platform bottom carriage is used, it must be
either bolted to a concrete foundation as in fixed
mounts or else it must have a vertical projection
similar to a spade on a field carriage to take up
the horizontal reaction in firing. Further, the
bearing surfaces of this platform must be suf-
ficient to prevent overturning of the carriage
firing at low angles of elevation or change in
level in firing at any elevation. That is, the
center of pressure of the reaction of the earth
must be within the middle third of the length or
diameter of the platform in the line of fire.
Since platform mounts vary considerably in con-
struction of detail no attempt will be made to
catalogue the various types used.

With fixed mounts the bottom carriage or plat-
form is usually secured to a concrete foundation
by a distribution of bolts along a circular flange;
and since with fixed mounts all round traverse is
possible, each bolt should be designed for maximum
tension.

21

DC

22

CATERPILLAR MOUNTS. To increase mobility

during the World War, cater-
pillar mounts were developed
extensively. A caterpillar
mount consists of an ordinary

gun mount including the tipping parts and top car-
riage mounted on a bottom carriage which fits with-
in the frame of the caterpillar. The caterpillar
is propelled by its own engine, and traverse can be
readily made by keeping one of the caterpillar
tracks stationary and moving the other. For more
delicate traversing the top carriage is provided
with limited traverse about the bottom carriage.
The essential features of the caterpillar proper
are:

(1) The frame which supports the
bottom carriage and the principal
bearings for the driving mechanism.
The caterpillar frame in turn is
generally supported on a series of
roller trucks which travel on the
caterpillar tracks.

Between the roller trucks and caterpillar
frame, spring supports are usually provided, and
the roller trucks are built to have more or less
up and down movement at their ends to conform with
the contour of the ground.

The frame may be either a casting or built up
of structural steel. The structural steel frame is
perhaps lighter but more subject to objectional de-
flections . .,»

The reactions on the frame consist of the
various spring supports from the supporting roller
trucks, the reactions of the bearings of the
running gear and the reactions of the gun mount
transmitted by the bottom carriage to the frame on
firing.

23

The frame of a caterpillar is subjected
to a complicated system of stresses. Due to various
possible loading conditions during traveling such
as the entire weight of the caterpillar being
carried in the center or else at the ends, we have
different types of loading reactions. Further a
wrenching action with corresponding large transverse,
stresses are induced by the supporting reactions
being on either side at the further extremities of
the track. This requires considerable lateral
bracing. In fact outside of fabrication and con-
struction considerations, the design of the cater-
pillar frame should be based on a careful analysis
of the various types of supporting reaction com-
binations that may take place in the traveling of

the caterpillar. It will be usually found that the
traveling stresses are somewhat greater than the
firing stresses and are often of an opposite
character.

The driving mechanism of the caterpillar con-
sists of two tracks each consisting of a continuous
track or belt of linked shoes. The caterpillar
track is driven, by sprockets usually at the rear
end. The drive shaft contains at one end the track
sprocket, and at the other end the drive sprocket
gear, which meshes by a suitable gearing to a
clutch, the system of gearing &nd clutch being
symmetrically the same for either track. The
clutches are driven by bevel gears or other
forms of reduction gearing through a gear box,
and sometimes a master clutch, to the engine
crank shaft. The traction gearing is straight
forward and is very similar to other types of
drive gear transmission. Mechanical steering
is obtained by operating either the right or
left track, holding it stationary or sometimes
reversing the motion and running the track
backward.

24

Electric drive caterpillar mounts are in two
(2) units and possess certain advantages'; first,
the transmission can be greatly reduced in either
unit by the use of compact motors a_nd gearing';
second, the units can be made similar and the
mobility thereby increased; third, a better design cf
gun mount is possible due to less limitations on
clearance and other corresponding factors. The
electric drive consists of the gun mount unit and
the power plant unit. The power plant unit sup-
plies power for driving itself, as well as the
gun mount; fourth, the caterpillar is braked in
traveling by suitable band brakes in the trans-
mission. When, however, the gun is fired, it is
necessary to brake the caterpillar from running
back. The braking and torque being usually in an
opposite direction and necessarily of a large
value as compared with the traveling braking1; it
is usually customary to introduce a band brake on
the final drive shaft and thus eliminate the
stresses in the transmission during firing. The
braking should be designed to produce a traction
reaction equal to approximately 80 percent of the
total caterpillar. Fifth, in a design of
caterpillar mounts, stability is of prime im-
portance due to the limited wheel base and
necessity of maintaining as light a mount as
possible. Stability may be increased by the use
of outriggers attached to the caterpillar body.
To decrease the overturning reaction of the recoil
on firing and thus increase the stability, double
recoil systems have been successfully introduced on
larger caterpillar guns. A double recoil system consists
of an ordinary recoil system between the gun and
cradle of the top carriage and a lower recoil
system between the top carriage and frame. The
top carriage is designed to roll up an inclined plane
of sufficient elevation to bring the recoiling
masses into battery and the cate-rpillar lies in a

r

k

tf

26

horizontal plane. This elevation is usually at
from 6 to 7 degrees. By the use of double re-
coil systems fhe stability is greatly enhanced, since
the inertia resistance of the top carriage creates
a stabilizing moment which is added to the
inertia resistance of the upper recoiling parts.
In the design of the double recoil system cater-
pillar mount, it is highly desirable that the top
carriage recoil as far as possible up the inclined
plane. Due to less limitations and clearance, an
electric drive of the two supporting units offers
a very suitable gun mount and a long recoil of the
lower recoil system is usually possible.

Fig. 6

RAILWAY MOUNTS Railway mounts developed

during the late war consist of
three (3) systems: (1) these
where the car mounted on suitable
tracks, rolls back on firing;(2).
those sliding back on a special track the tracks
being disengaged, (3) platform or stationary
railway mounts with suitable outriggers, the

27

trucks being entirely disengaged. In types (1)

and (2) a very limited traverse is possible,
whereas, in type (3) considerable amount of traverse
is possible.

Railway mounts of type (1), rest upon suitable
girders, supported by the trucks at either end. The
girder must be designed to carry the maximum firing
load stresses at maximum elevation, as well as
stresses due to the dead load weights. The trucks
take the supporting reactions from the girders of
the dead weight load as well as fhe firing load at
maximum elevation. Great care is needed in dis-
tributing the loading from the various axles by
properly formed truck equalizers.

In type (2) a special built-up track is
necessary, the trucks being disengaged merely
carrying the dead weight of the mount. The mount
is designed to have a considerable bearing surface,
and thereby the bearing pressures are greatly reduced

In sliding railway types, recoil systems have
in certain types been completely eliminated, the
recoil being merely resisted by the friction of
the track. Due, however, to the enormous stresses
due to high caliber guns at maximum elevation,
recoil systems should always be introduced.

With stationary or platform mounts the
question of stabilizers of corresponding outriggers
become a fundamental feature in this type of design.
Platform railway mounts have similar characteristics
as ordinary field platform mounts in mobile
artillery.

28

rig. 9

Fig. 10

CHAPTER II

DYNAMICS OF INTERIOR BALLISTICS AS AFFECTING RECOIL
DESIGN.

The object of interior ballistics is partly to
derive expressions for the acceleration and velocity
of the projectile during the travel in the bore, and
the corresponding pressures on the base of the shell
and breech in terms of tne powder loading, the form
of powder grain, the initial volume of powder chamber
in the gun, and other variables upon which the
velocity and pressure depend. In the design of the
recoil mechanism as well as the carriage for its
maximum stresses, it is very important to know the
accelerations, velocities, and pressures in the gun
to a considerable degree of accuracy throughout the
time the powder gases act.

In the study of interior ballistics, it is con-
venient to divide the powder pressure interval into
two periods:

(1) The interior period while the shot
travels up the bore to the muzzle.

(2) The after effect period while the
powder gases expand after' the shot has
left the muzzle.

During the interior period, we have considerable
combustion of the charge and corresponding gas evolved
in the powder chamber before the shot has left its
initial position in the breech end of the bore, the
temperature rising and the pressure reaching a
value sufficient to force the projectile into the
rifling groove and to overcome initial frictions,
usually a considerable fraction of the powder pres-
sure obtained. The projectile then moves up the
bore followed by further combustion and expansion
of the gases evolved from the combustion of the
powder. The combustion exceeds the expansion up

29

30

to the tine of maximum powder pressure which is
reached after a travel up the bore roughly from
1/6 to 1/3 the length of the bore depending greatly
on the type of cannon, charge, etc.

The energy of combustion is expended:

(a) In Kinetic Energy of translation
? of the projectile.

(b) In Kinetic Energy of translation
of the recoiling mass (assuming the
recoiling mass free).

(c) In the Kinetic Energy of the
charge itself.

(d) In the work on the rifling and in
friction.

(e) In the angular energy given to the
projectile.

(f) In dissipated heat.

The last three are very small as compared with
(a), (b) and (c). Further (b) and (c) are small as
compared with (a).

"Ingalls" states that about 83* of the total
energy of the work of expansion goes into the
Kinetic Energy of translation of the shot, the re-
mainder 17* going into the forms b, c, d, e and f.

The rate of combustion depends upon the forn
and size of the grain, it being an observed fact
that powder burns in layers always parallel to the
initial surface. Further the rate of combustion is
a function of the actual pressure generated, vary-
ing as some power of the pressure. The value used
for this exponent is one of the most tentative
features in the whole subject of interior ballistics.

DYNAMIC RELATION- Let IT = the mass of the pro-
jectile.
* = the weight of the

SHIPS IN INTERIOR

BALLISTICS.

projecti le .
m = the mass of the

charge

31

it = the weight of the charge.

mr= the mass of the recoiling parts.

wr= the weight of the recoiling parts.

u = the travel up the bore.

x = the absolute displacement of the shot in the

bore .
X = the corresponding displacement of the recoiling

parts .

v = the absolute velocity of the shot in the bore.
vo= the muzzle velocity of* the shot.

V = the free velocity of the recoiling parts

(absolute ).
Pxj= the total pressure on the breech.

P = the total pressure on the base of the shot.
Pk= the intensity of pressure on the breech

(Ibs. per sq . in. ) .
p = the intensity of pressure on. the base of the

shot (sq. in.),
f = the component of the rifling reaction parallel

to the axis of the bore.

Then,

dv
P - f = m -— , for the motion of the

d t,

projectile

dV
and P^ - f = mr — , for the motion of the

recoiling mass in free
recoil (2)

and further assuming the charge to expand in
parallel laminae with the successive laminae
having velocities as a linear function of the
end velocities, we have,

p. _ p = 5 /dv dV .

D 9 V - — ) f n\

2 dt dt (3>

where

dv dV

dt dt = the mean acceleration

32

of the powder.

Combining (l), (2) and (3)

I , dV ffi x dv

<*<• * — > at = (n + T ? at (4)

Integrating successively,

(m * JL) V ••<• + 4-)v (5)

(mr * -1-) x - (» + ~)x (6)

c £

The absolute displacement of the shot in the
bore is connected with the travel (u) up the bore
by the following relation:

x = u - X

since the positive value of X is assumed opposite
to x.

Substituting in (6), we have,

i .

(m + — - ) u
X - 2_ (7)

mr+ m •*• m

which gives the relation of free recoil to the
travel of the shot up the bore.

Obviously (5), (6) and (7) may be written
immediately from the principle of "linear
momentum" (that is, the total momentum of the
system remains constant unless acted on by external
forces) and the principle that tha center of
gravity remains fixed unless acted upon by external
forces. In free recoil the exterior forces are
nil.

The pressure on the breech exceeds that on the
shot by the inertia resistance offered by the
of the powder gases,

33

>nrdV
mdv

Neglecting ^ as small compared with mr,
2

pb - f (m

P - f
hence

Since the rifling reaction expecially during
the movement of the shot up the bore is roughly 2
per cent or less of the value of p, we may entire

ly neglect the term f .

'

in the above expression, which simplifies to

ft
m + —

Pb = - 2P (8)

•

From a series of experiments conducted by
the United States Navy the value

E
m + - -

2m
- = 1.12 a constant, approx.

•

hence

pb- = 1.12 F approx. (9)

It is to be noted that the acceleration of
the powder is very likely somewhat different

from the assumption upon which (8) was de-
rived, but nevertheless equations (8) and (9) give
a good approximation of the increase of breech
pressure over that at the base of the projectile.
During the "forcing in of the rifling" before
the commencement of motion of the shot, obviously
Pb = P-

34

According to the previous assumptions the pres
sure varies progressively, decreasing from its
maximum value at the breech block to a slightly
smaller value at the base of the projectile.

Therefore, if we let p^ be the average or mean
instantaneous pressure or rather the pressure in
the -powder chamber and bore, we have, p + p

• P = -—

*» 2

In terms of the total pressure at the base of the

projectile,

m dV dv m

p , . 7T . ,21 -3T, dv

m * • 2^

dt

but dv

hence

-•_ = p=p(i + ^_) = p(1+ _L_) (10)

» 4 in Aw

or in terras of the total breech pressure

dV "rar dV
mr dt + ^»_ ~tt

Pm « 5 ~ (1

m
" * 2

(11)

ID * 1

2

35

EQUIVALENT The riflil1g grooves in

MASS OF the gun come in contact with

PROJECTILE the copper rifling band on the
projectile and angular motion
is transmitted to the projectile

in addition to the translatory motion. The object
of the angular motion is to give the projectile a
gyroscopic effect maintaining, with a combination
of the air reaction, the axis of the projectile
parallel to the tangent of the trajectory and
further making an oblong projectile possible with
greater ballistic efficiency.

Let
P = the reaction of the powder on the base of the

shell.

m = the mass of the projectile,
f = the total rifling reaction normal to the rifling

groove,
uf = the friction component of the rifling reaction

tangent to the rifling groove.
6 = the angle of pitch of the rifling, (i. e. the

angle the rifling makes with the axis of the

bore ).

p = the pitch of the rifling
d = the diameter of the bore
k = the radius of gyration of the projectile..
x = the displacement of the projectile up the bore

from its initial position.
£5 = the corresponding angular displacement twist of

the projectile.

Then we have,

P - f(sin e 4 u cos 9 ) = m 4!*-

dt \i£>)

f(cos e - u sin 9 ) | = mk' £f (^}

36

Further since, the number of complete turns
or revolutions of the projectile in its linear
displacement x or its angular displacement ft,

is

JL or JL
p 2*

i»e have

da# = 2* dax (14)

ar*" ~p~ "31"*""

In terms of the angle of pitch of the
rifling,

- 16 = x tan e or n tan e

= ""

hence

2 d*x

T tan 6 dF~ (is)

Substituting (14) or (15) in equation (13)
we have

* dax d*x

mk* 4 "P" dT» mkMtan

--- — -------- = - -

(cos 9 - u sin e)d (cos 9 - u sin e)da

(16 )

which shows the reaction f is always proportional
to the linear acceleration of the projectile.
Therefore, the friction uf , is also proportional
to the linear acceleration.

Substituting (16) in (12), we have

or in terms of the rifling angle,

.sin e * u cos e \ 4k*tan@. d*x

P - I 1 ^cos 6 - u 51B b > ~li - J ffi ip

37

which shows that the powder reaction P is also
directly proportional to the linear acceleration
of the projectile. Evidently the equivalent mass
of the projectile, is

.sin e «• i

i cos 6 <

cos e - i

i sin 6 j

.sin Q + i

i cos 6 >

^cos e - '

j sin e ;

dp

4 tan e ^a -, 0\
"

Hence the rifling reaction and friction due to
rifling are directly proportional to the powder re-
action, that is the pressure on the rifling grooves
always varies at any instant directly with the powder
reaction.

fbus we have the relationship that rifling.
frlc.tj.on behaves exactly like, an additional mass:
that jLaf it has an inertia^ef f ect since it is_prft-
portional to the aeeeleratipn.

The true equivalent mass due to the linear and
angular inertia of the projectile alone, can be ob-
tained by assuming the rifling friction zero, ( i.e.,
putting u = o)

4 n8k*v
n' • (1 — - - ) m

(i " ::" ° ) . (20)

The true equivalent mass may be readily checked by

a consideration of the total energy of the projectile,.

that is,

\ m'v2 « i mva + y In*

2nv 2v tan

and w • - -

38

and I = mka where k = radius of gyration about

its longitudinal axis.

hence . » . *

4n k . ,, 4k* tan* 9 N

B1 - ( 1 + = — ) m = (1 + — ) m

D* d*

EQUIVALENT MASS For a differential layer of
OF the powder charge at the base of

POWDER CHARGE the projectile, its velocity
evidently is equal to that of
the projectile while for a dif-
ferential layer at the breech, the velocity is equal
to that of the gun. For intermediate layers, we must
assume some law of variation of velocities, between
the two end limits. For simplicity and probably
a fairly close approximation, we will assume for the
various laminae, a linear variation of velocity be-
tween the end limits. Further since the velocity
of the gun is small as compared with that of the
projectile, in virtue of the approximation of the
whole analysis, we are entirely justified in assum-
ing the recoil velocity entirely negligible.
If,

Velocity of projectile * v (ft. sec.)

Distance between breech

and base of projectile = x (ft.)

Velocity of any inter -

mediate lamina = v1 (ft/sec.)

Distance from breech to

lamina = x1 (ft.)

Then

v1 = — v (ft. sec.)
u

If we assume the density of the powder is
uniform through the distance x, so that the
weight of the lanina is x dx, then the kinetic

39

energy of the lanina is

W v ' W V 2 o

V 1 I " * I £• * •

TT o dx or — . -7T * *X

2 g x» 2 g

and the Kinetic energy for the total charge
becomes,

K. E. of w - f. _Ii rx x'2 dx1
x» 2g ^o

' J (i. ) T- (21)

That is, the equivalent mass, when dealing

with the energy equation, is 1/3 the mass of the
total charge.

It is important to note that when dealing with
momentum, the momentum for the total charge becomes,
on the same assumption

w u x1 w
JL / JLvdx'=.JLv (22)
gx o x 2g

that is the equivalent mass from the moment or
aspect is 1/2 the mass of the total charge.

EQUIVALENT MASS It is convenient in deriv-

OF ing the energy equation,

THE RECOILING PARTS to express the Kinetic

Energy of the recoiling parts
in terms of the velocity

of the projectile.

Neglecting m ds small as compared with mr
2

and, neglecting the recoil brake reaction as small,
we have, by the principle of linear momentum,

f ™ \ ( \
mr V = (m + - ) v ( approx.)

40

hence (m * )

Therecoil Energy, becomes,

? »r v* = i I va (23)

and therefore the equivalent mass of the recoil
ing parts, in terms of the velocity of the
projectile, becomes,

(m + JL )«
2

ENERGY EQUATION The mechanical work

expended by the gases of the
powder charge in the bore is
equal to the external work ex-
erted on the projectile and

gun, plus the Kinetic Energy given to the gases
themselves, plus the heat energy lost in
radiation through the walls of the gun.
If

W = the Potential Energy of the Gases at any

instant.
P|,j= the total reaction exerted on the breech

of the gun.
P = the total reaction exerted on the base of

the projectile.
X = the displacement of the gun measured in the

direction of its movement.
x = the displacement of the projectile measured

in the direction of its motion.
E = the Kinetic Energy of the powder charge.
Q = the loss of heat due to radiation.

41

J » the mechanical equivalent of heat = 778 *** lb*

B. T . U ,

Then for the energy equation of the powder gases, ire
have,

-Pb dX -P dx * d(E + W) * JdO (1)
hence

-dW » Pb dX + P dx + dE + JdQ. (2)
that is the loss of the potential energy of the gases,
due to a differential expansion goes into mechanical
work (PjjdX + P dx + dE) and radiation JdO.
Further by (19), (23) and (21),

P dx = d[y(m"va))

pbax

AS = tlf (5 v» )]

0

so that

Further, in terms, of a hypothetical mean
pressure Pm(over the cross section of the bore)

equation (3) may be expressed in terms of the
travel up the bore u, (i.e. the relative displace-
ment between the gun and projectile).

where

(6)

(6)

jec
the bore of the gun, approximately since

dv

where v --— = the acceleration of the projectile up
du

42

dv dx dv dv

l~

dx

v dV = df dx + dx = d? l~TdT> and dX is

pared with dx, and

m" = the equivalent mass of the projectile
which takes care of its angular
acceleration as well as the rifling
friction, see equation (19).

EXPANSION It will be assumed, that the

OF expansion of the gases due to

POWDER GASES the combustion of the powder

charge obeys the law of a perfect

gas. Hence, we have,

PV = RwT
where p = the Intensity of Pressure exerted by the

gas Ibs/sq. ft.

V = the volume of the gas. (cu. ft.)
w = the weight of gas (Ib.)
R = a coefficient (ft. Ibs. per Ib . gas.)
T = absolute temperature reached.
Further, with a perfect gas, the internal
energy of the molecules of the gas is entirely in
a Kinetic or Vibratory form, and therefore, is
directly proportional to the temperature.
Hence, we have,

dQ
dQ = cwd T and c = —

where dQ = the heat required to raise the gas for

a change of temperature dT.
c = Specific heat or the heat required to
raise one Ib . of gas one degree of
temperature at the temperature considered.
We are concerned especially with the expansion of
a gas at constant pressure or at constant volume or a
combination of the two.

43

pdV
Hence, dQ. = wCp dT = Cp =-- — at constant pressure

Cv

dQ » wC dT =

R at constant volume.

If the volume and pressure vary together, then,

we have the sum of the partial variations, above,
that is,

dQ »—- (C p dV + Cv V dp)

Rw "

but since, dT = — (p dV + V dp)
R

we have, dQ = CywdT + °P ' °v p dV
and Q _£

a = wCv/dT + p R v — /p dv

This relation can be interpreted, physically
immediately, since the internal energy being entirely
of a kinetic or vibratory form, must be proportional
to the change in temperature at constant volume other-
wise additional heat must be added for the external
work. Hence wCy/dT measures the molecular kinetic
energy. Considering an expansion at constant pressure,
the total heat required is,

Q = V 4-pdV

•I

where U * the internal energy = wCy dT.

Since the heat is added at constant pressure,
we a Iso have,

pdV = wR dT

But the heat added at constant pressure is,
Q » wCp dT
hence, substituting in the total heat equation,

wCp dT = wCv dT — w R dT

^Dd R _R _
Cp - Cv ' T- or J -

P - <-v

44

If now the specific heats Cv and Cp are assumed
constant for the range of temperatures during the ex-
pansion of the powder gases, we have,

Cp - Cv
Q = *CV(T - TJ + -^ W (7)

where

If is the external work performed.

Tt is the initial temperature.

Neglecting the loss of heat by radiation as small,
we have practically an adiabatic expansion in the bore
of a gun; that is,

- C,

wRT
Since p = -=-, dividing by T, we have

Cv " + <Cp - Cv) |L 0

and if we let

Cp

— = " **«" j X" ./v

dV

Cp <jx
-i- = n, then — + (n-1)^- = 0
uv
and ,

It ' V
Therefore,

T- . c^)"-1

Ta V ;

Now for an adiabatic expansion, Q. = 0 and therefore,
(eq.7) becomes,

RwCv(Tt - T) T,

» = -7; « — (Tt - TJ

Cp - Cy n -

hence

7 n-1

= wCv J Tt [1 - (•—-) 1 (8)

This equation 1* in convenient form since it it
in terms of the initial and final volume in the bore.

The equivalent length of the powder chamber of
the gun in terms of the area of the bore, becomes,

V,

Vi uo

0.786 d* so that

V u0+ u

In terns of the displacement up the bore the
work of the adiabatic expansion of the gases, becomes,

W » wCv J T,. f 1 - ( - — )" " l] (ft - Ibs)

1 (ft - Ibs.) )

) (10)

*

i (- • )
n - uo * u

Here

w» weight of gases (Ibs.)

W « external work performed during the

(adiabatic) expansion (ft. Ibs.)
Cv* specific heat for constant volume (B- T. U. per

Ib. per deg. )
Cp= specific heat for constant pressure (B. T. U.

per Ib. per deg.)

T,= initial temperature "(degrees)

Vt» initial volume (i.e. volume of powder

chamber)

V«.

u0 = _± - where 0.7854d* • area of bore
0.7854d'«

u = displacement of projectile up the bore
J = mechanical equivalent of heat (• 778 ft. Ib.
per B. T. IT.)

46

R» (C CV)J= the gas constant (ft. per degree)

From this equation we may deduce the differential
equations of velocity and powder pressure for the
movement of the projectile up the bore, provided
we know the manner of burning of the powder gases,
etc.

The energy equation therefore becomes:-

2 wRT* (i . .!£.. )n - l = ." ! 1 • v2 11

' n - 1 u0* u ; mr 3 J (11)

From which we may determine v in teras of u.
The factor dg which represents the loss due to

beat radiation must be determined by experiment.

Based on the energy equation (or its
derivation, the force equation of the motion of
the projectile in terms of the displacement up the
bore) various interior ballistic formulae have been
derived differing in the method assumed as to the
combustion and expansion of the charge. The
formulae of Ingalls and Hugoniot have been used
by our Ordnance from time to time especially in
ballistic calculations. In recoil design, however,
rough approximations are sufficient since the
manner of combustion has small effect on the
recoil. The formula of Leduc is sufficiently
condensed with sufficient approximation to be
admirably suited for recoil design.

TOROUE REACTION It is important in

OF the design of traversing

THE PROJECTILE gear for guns shooting at

high angles of elevation to
compute the average torque
reaction of the projectile upon the gun.
Let

w s ang. vel. of projectile at any point in the
bore (rad/sec)

47

v = linear velocity in bore along X axis (ft/sec)
r = radius of bore or of projectile (ft)
Pjj = powder pressure at base of projectile (ibs)

f = normal reaction of rifling groove (Ibs)

T = torque on projectile (Ib. ft.)
6 = angle of rifling grooves with XX
I = mk* = moment of Inertia of Projectile
IS = angle turned by projectile
Then

u = v tan e

d*0 = tan e dgx (21)
dt* r dt2

hence

, d 0 mk* „ d*x

T = nk* = — tan 0 — - , ^.

dt* r dt* (12)

but

Tt2" m (approx.)

hence T = ~ tan 0 Pb (Ib ft.) (13)

2*r
where tan 0 =

p being the pitch of the rifling at the point
considered .

From equation (23) we see that the torque is
proportional to the powder reaction on the projectile,
and the "slope" of the rifling grooves, the steeper
the grooves being the greater the torque reaction
with a given powder pressure.

Further, if the rifling pitch is made constant
throughout the greater part of the bore, the torque

48

varies as the powder pressure curve along the
bore and therefore is a maximum at the beginning
of the travel of the projectile.

For the average torque, we have,

d»0
Tav s »k* ( - ~\v

u t

The moment of inertia of the projectile
may be roughly evaluated by assuming a solid
cylindrical projectile:-

If C * mean or equivalent length

D = density or weight per cu. ft.

DC r DC

- ' 2 n r dr = ~

DC
but m * -—

hence -E£-nr*k-* -— -S~ and k« «-^- ;
g 2

k - 0.707 r

Further, it is customary to designate the
"rifling" as a "twist" of 1 turn in "g" calibers.
Therefore, if w,e let,
Twist: = 1 turn in "g" calibers

Time of travel in bore (approx.) 3 t » 3/2 -

v

Radius of gyration of projectile » k * 0.7 r

Number of rev. per sec. = n

Then,

n g 2 r = v )

an£ ) at the nuzzle

d0 HV )

—— * 2nn = -

dt gf d0 }
Therefore, since T...t = mk(r — ) ,

dt m

T -r— - mx 0.49 r* -
2v gr

and

rev "

T » i.05---r (24)

ug

49

which gives the average torque reaction on the gun due
to the angular acceleration of the projectile.

Since the slope of the rifling grooves is small,
we may roughly assume, that, a

fr = T and f = 1.05 - (35)

ug

which gives the mean pressure on the rifling band.

It is of interest to compare the maximum torque
reaction to the average torque reaction in an actual
gun.

Type of Gun: 240 m/m Howitzer

Muzzle Velocity: 1700 ft/sec.

Weight of Projectile: 356 Ibs.

Max. Powder Pressure: 32000 Ibs/sq.in.

Rifling = 1 turn in 20 cals.

Travel up bore = 13.33 ft.
Then, for the max. powder reaction, we have,

Pbmax. = 32000 x 0.7854 * - = 2,242,000 Ibs.
and for the rifling slope,

hence, for the max. torque reaction,

9 45
T = _: — x 0.157 x 2,242,000 = 69,500 Ibs. ft.

48
where as for average torque, we have,

u

1.05 x 356 x 1700 x 9.45

T „ „ = - — _ = 49600 Ib. ft.
av 13.33 x 32.2 x 20 x 24

Therefore, the ratio of max. torque to tne
average torque becomes,

Tmax _ 69500
f av " 49600 =

Due to the short time action of the travel up the
bore, the effect on the traversing gear depends upon the
average torque rather than the maximum.

50

MUZZLE BRAKE

GENERAL The muzzle brake consists of curved

DESCRIPTION vanes secured to the end of the muzzle
upon which a portion of the powder

gases are deflected in the second part
of the powder period after the projectile
has left the muzzle. The gases are deflected somewhat
to the rear, and we have a forward reaction due to the
change of momentum of the gases, which materially checks
the recoil. The design and best arrangement of vanes
requires a considerable experimental investigation and
it is merely proposed here to outline certain general
limitations based on an elemantary theory.

ELEKANTARY If it were possible to calculate
THEORY the mass of gas discharged through the
vanes, as well as the mean extrance and

exit velocities, the reaction on the
vanes could be determined. But the

•etbod is complicated, since the pcmder pressure
after the shot has left the muzzle falls off accord-
ing to a complicated function of the time, and, con-
sidering the variable volume of gas, this makes it dif-
ficult to approximate the mass of the gas as a function
of the time. Further the amount discharged through the
vanes depends upon the initial mean muzzle velocity of
the gases, the caliber of the fcore and the entrance
areas to the vanes, as well as the variation of muzzle
velocity of the gases against time. We see therefore
to approximate roughly the problem from a theoretical
point of view would require an elaborate analysis com-
bined with a long and eleborate experimental
research.

Let

w » weight of projectile (Ibs)

5 * weight of total charge (Ibs)
v » nuzzle velocity ft/sec.

51

Vo = velocity of recoiling parts when the

projectile leaves the muzzle ft/sec.
to = time for shot to reach muzzle.

T = total time of powder period.
Pjj = total pressure on breech due to powder gases

(Ibs)
Wr = weight of reeoiling parts

vw = mean velocity of gases after free expansion

(ft/sec. )

T
/ Hdt = impulsive reaction on vanes (Ibs)

„ charge through vanes

Cw = ratio of s — - — r5

total charge

Then, without vanes, we have,

pb dt = ° Pb dt +{pbdt Total
o t0

reaction on the

Gun during the
(w + - ) v Powder period.

Now / ° Pb dt =

o g

and since the powder charge has a mean velocity =

v

— — when the projectile leaves the bore,

H

± W

hence / pb dt = v + - vw
o £ g

therefore, WrVm = wv * w vw

as we should expect from the principle of the con-
servation of momentum.

With the muzzle brake acting, the total reaction

52

during the second period (T -t0), on the gun becomes,

/T Pb dt - ;T8 dt
'

and therefore the momentum imparted to the gun be-

wr (v V \
dt » - <vra - o '

comes,.? _ T wr (v V
P dt -

o

If it were possible to deflect the total charge

entirely backward and maintaining the same expansion,
then, for the total reaction on the gun during the
expansion period of the powder gases, we have,

V

since the change in velocity * vw + - ft/sec

ft

Therefore, the momentum given to the gun during the
powder period becomes,

wv - w v_.

which gives the impulse imparted if we had muzzle
brake, with the sane expansion backward as forward
through the vanes.

With the same expansion to the rear the
maximum possible recoil energy that can be absorbed
with a muzzle brake, becomes,

Aab

2 w w v vw

* »r

and the maximum possible percentage of the total recoil
energy absorbed by the ideal muzzle brake becomes,

53

Aab 4 w w v vm

A (wv + w vw)*

Since w vw is always less than wv, we see that
even with an ideal muzzle brake and complete expansion,
it is impossible to completely check the recoil energy,
unless greater expansion is obtained to the rear than
forward.

The total gas reaction on the gun due to the com-
bined expansion and deflection of the gases, is
represented by the impulsive reaction,

T T

/ R dt -/ Pjj dt in a forward direction (i.e.
to to towards the muzzle)

If we have complete expansion of the gases,
before entrance into the muzzle vanes, then,

;T pb dt - ^ (v . y. )

«* '0 g ^ W U

and if now the total gases are deflected entirely back,

then

T 2 w v..

r R dt = - •

*

and as before,

If, however, the gases are accelerated to a
mean velocity v1 before entrance into the vanes,
then

1* V

/T Pb dt - - (v- _ ' )
g

and further expansion takes place through the vanes
to the maximum value vw to the rear, we have

/ R dt = - (v« + v )

since v1 + vw is the change in velocity.

As before the total impulsive reaction on the gun,
becomes,

£H at -£ pb at .j (,„ , J>

Without the vanes, the reaction on the gun breech be-
comes,- R v

{I ?* « • ; K - I )

and with the vanes the reaction on the breech is pro-
bably different and modified to,

/T Pfc dt = * ( V' - J)

to

since some expansion probably takes place within the
vanes, themselves.

Now as to the actual reaction obtained, the
ideal brake differs from actual conditions, essentially
in the following points:-

(1) Only a part of the total charge can
be deflected through the vanes.

(2) The entrance velocity can only be a
component of the actual muzzle velocity
of the gases.

(3) Only a partial expansion of the gases
can take place before entrance into the
vanes .

(4) The exit velocity can not, for
practical considerations, be entirely to
the rear, 30* from the rear, being like-
ly the maximum angle that the gases can
be deflected.

(5) Only a very small expansion can take
place through the vanes themselves; of
the gases passing through the vanes the
total expansion is small.

In consideration of (1) (unless the vanes are ex-
tended a considerable way out) the higher the muzzle

55

velocity the less the total charge passing through the
vanes. It has been found experimentally that it is
useless to add more than a given column of vanes,
further addition of vanes having very little effect
on the reaction. Further the first one or two vanes
nearest the muzzle, are subjected to an intensity of
pressure practically equal to that of the gases at the
muzzle. Further development of the muzzle brake should
be directed in obtaining greater expansion to the rear
by a suitable combination of vanes, curvatures of same,
etc.

LEDUC'S FORMULA The empirical formula estab-
lished by Leduc is especially service-
able and sufficiently accurate for a
predetermination of the reaction of
the powder, during the powder period
and its effect on the recoil.

Leduc's formula, assumes that the velocity curve
of the projectile during its travel up the bore follows
that of an equilateral hyperbola, with parameters a and
b, that is,

If v = the velocity of the projectile at any point

in the bore (ft/sec)

u = the corresponding travel up the bore (ft)
a and b being parameters of the hyperbola,
then

v = r-r: — (ft/sec)

b + u

where a and b must be determined by the elemantary
principles of Interior Ballistics.

Determination of the parameters a and b:-

When u is made infinite, that is u = a

and v - a - a

then a

56

a is therefore determined by considering the expansion
in a gun of an infinite length. c

If n » the ratio of the heat capacities (—*- *1.4l)
and for an adiabatic expansion pV = k, v

Then the work of an expansion from, initial Volume
Vt to final Volume V , becomes,

i
= / p d V, but p = — * where k and n are con-

Now when Vf becomes infinite

W * — — * t (ft. Its.)
n - 1 Vt

Since. 1 Ib. of water * 27.68 cu. in. for unit density,
k 1

i - 1 27.68" Expansion at Unit

Work for an Infinite

Expansio

Density.

Weight of given volume of powder gas
Weight of same volume of water
and if

7C » the given volume of the chamber (cu. in.)

Vt » the volume of 1 Ib . of gas (cu. in.)
then ^

** 27.68

A * » — - — per Ib. of powder gas.

*a

27.68

hence the specific volume of the gas, becomes,

27.68
V*a ' "I"

57

Therefore, the work of expansion of 1 Ibs. of the
gas to oc becomes,

\f An~l

W

1 27.68°

Since the gas evolved is proportional to the weight
of the charge w, and a = v for an infinite expansion
in the bore, we have

waf

w E A * ^ £or a compiete expansion of w

2g

Ibs. of powder gas,

hence a » /2gE (-*—)* A — 2

Now S has a value = 653 ft. tons roughly, and by
experiment "_lJ. s 1/12 (approx.) Taking into account

the various losses, it has been further found ex-
perimentally that /2gE = 6823 for ordinary good
powder.

Therefore, the parameter "a" becomes,

a * 6823 (— )t ATI ttf

W

27.68
Now A a -n

but with a powder chamber Vc, loaded with w Ibs.
of powder, the specific volume of 1 Ibs. of powder
evidently becomes,

t^
''ta * w assuming complete combustion of the

charge,

(2)

that is the density of loading may be defined as the
ratio of the weight of the charge to the weight of a

58

volume of water sufficient to fill the powder chamber,

Hence the parameter a becomes,

i __i
a = 6823 (-£-y ,27.68 w.ta

H f \ y /

vc

To evaluate the parameter b, we must consider the ac-
celeration of the projectile, and the reaction of the
ponder gases on its base, during its travel up the
bore.

The acceleration up the bore, becomes,

jiv (b * u )a- av K a2 bu
V"=

(b + u )' (b * u)3 (3)

since

3. V

v = fr4.u from Leduc's formula, hence the
pressure against the projectile, for a displacement
v, becomes,

w aabu
P = —

g (b+u)2 (Ibs)

Further the maximum pressure occurs, when T—
i.e. when,

] .4 .3
= -3u(b + u) + (b+u)

(b+u)4 and u = - (ft)

that is the maximum pressure in the bore occurs at a
displacement equal to one half the parameter b or the
parameter b - twice the displacement of the maximum
powder reaction in the bore. We have, therefore,

substituting v = ° in (3)
2

59

P ' - (Ibs) (4)

The mean powder reaction on the base of the
projectile during its travel up the bore, becomes,

where

VQ = the muzzle velocity ft/sec.

UQ = the total travel up the bore (ft)
The pressure against the projectile when the shot
is about to leave the muzzle, becomes,

a bu

Hence to determine the parameter "b" we have the
following equations:-

4 w a .
m 27 g D

f°-*

\ where Pm VQ and UQ are
* a*^_ ( known.

2

wv

and PQ Pe, a and b are
au
v : b+tl ( unknown.

)

Hence a solution is possible:- If A^ = Area of bore
and Pm = a given property of the powder used

Pm = 30,000 to 33,000 Ibs/sq.in. usually.

Pm = p^ A for the max. powder reaction.
Substituting atl

e 2g (b+u0)2

o

2

(Ibs) hence a =

60

P . 4 » £ (its) bence a"
• 27 g b

4 w
Equating, we have,

2 Pe(b * uo)» _ 27 b Pm

hence

27
b» + 2 b u0 * u*0 =

o

27

t

e

+ 12 - 3- — ) v b + u20 = 0

8 P.

Solving, we have,

(9 2? m 1 , S<9

"(2 " 8" Pi > "o ! /(2 ' g- F* • ' ' o -

(ft.) (7)

which determines the parameter b, in terms of the
travel up the bore, the given maximum powder reaction
and the mean powder reaction, being determined from
the muzzle velocity and travel up the bore.

To completely determine the velocity, powder
force, and time against the travel up the "bore, we
have

— aa- (ft/sec)

b + u

61

P =

g (b+u)3
and the corresponding time of travel, becomes,

, du (b+u)
* - / y- ' (au) du

b , 1
= a l°#e u * a u * Constant

Now when u = 0, t = 0 and loge u = - a, and the
constant cannot be evaluated without making some as-
sumption. Since the initial powder reaction required
to force the projectile into the rifling grooves is
large and the displacement u » « , to Max. powder
pressure is small, we can reasonably assume the powder
reaction constant and equal to the maximum- powder re-

action during the initial travel u = ^ . Hence as-

2
suming the maximum powder reaction to be reached at the

beginning of the travel of the shot up the bore, and

then

and substituting Praax> from

to remain constant up to u =

hence t. = v-~ (-) (8)

Sunstituting in the previous time equation, we
have,

'27" b b b b

(-) = — log - •»• — - * constant
a a 2 2a

62

and

Constant = — ((/27 - 1 ) -2 loge ^ ]

a\ (2.098 loge - )
2 "a

therefore

2u u

= - (2.3 log — + - +2) (approx.) (9)
a b b

The powder reaction on the breech during the
travel up the bore is somewhat greater than at the
base of the projectrle due to the inertia resistance
of the powder gases and charge. It has been shown
previously that the breech pressure is augmented
over that at the base of the projectile by either of
the two following formulae:-
v, + |

Pfe = p (Ibs)

w

or

Pb = 1.12 P (Ibs)

The former is based on a theoretical assumption,
and gives an idea as to the change in the pressure

drop from the breech to the projectile with different
ratios of powder charge to weight of projectile. The
latter is entirely empirical and it appears that the
ratio of the weight of the charge to that of the pro-
jectile has no effect on changing the ratio of the
breech pressure to that at the base of the projectile.
Unfortunately the latter empirical value is somewhat
limited especially for extreme ratio of the projectile
weights but is, however, reasonably accurate for
ordinary calculations. The former is more or less in
error due to the assumptions made, but it gives the

63

characteristics for extreme ratios. Therefore, for
extreme ratios of charge to projectile weights, the
former formula should be used, while with ordinary
ratios, the latter should be used.

R e c ap i t ul ait ion of the Various Formulae
Originating from LIDUC'S Formula -

Let

v = Velocity of projectile up "bore (ft/sec)

u = Travel up bore (ft)

v0 = Muzzle velocity (ft/sec)

UQ = Total travel up bore (ft)

t = Time of travel up bore (sec)

to = Tine of total travel up bore (sec)

W = Weight of powder charge (Ibs)

w = Weight of projectile (Ibs)

Vc = Volume of powder chamber (cu.in.)
A = Density of loading

P = Powder reaction on base of

projectile (Ibs)

P^ = Powder reaction on base of breech (Ibs)

PJH = Max. Powder reaction on projectile (Ibs)
Pe = Mean Powder reaction on projectile (Ibs)
A<j = Area of the bore (sq.in.)

pm = Max. given powder pressure (Ibs/sq.in.)
(from 30,000 to 33,000 Ibs/sq.in.)

Given :- Pm vo, Vc, w w
To evaluate: — v, P and t

64

27.68
then, A =

vc (i)

i = 6823 (-)7 Al/l* (2)

(Ibs) (3)

(Ibs) (4)

U° 16 pe 16 pe

v = -^— (ft/sec) <6)

b + u

P = -

g (b + u)» (Ibs) (7)

w a 2bu

(b+u)3 (Ibs) (8)

w a2bun

P0v = 1.12 - 2

)3 (Ibs) (9)

t = - (2.3 log ^ t - * 2) (sec) (10)

a bo

- (2.3 log ^ + ,
a bo

t0 = - (2.3 log + ,— +2) (sec)

3 ^
2 vo

v (approx.) Uec) (11)

65

DYNAMICS OF RECOIL The velocity and displace.

DURING THE TRAVEL OF ment of the recoiling mass

THE SHOT UP BORE with respect to the powder

charge and projectile is ob-
tained by the principle of

linear momentum.

Assuming, one half the charge to move forward with

the projectile and the other half to move backward

with the recoiling parts, we "have,

( «r * \ >Vf = ( » * \ )v

and (wr = - )Xf = ( w + -• )x
2 £

Now the absolute displacement of the shot in the bore
is related to the travel up the bore u, by the
equation

x = u - X
Hence, we have,

and

w
(w + - )u

Xf =

Since w and w are small as compared with wr, we have

for a sufficient approximation

w
w + -

Vf = 2- v (ft. sec)

w_

X -iLlJl

Af - ' *' U ft.

The equation of velocity displacement and time
of free recoil during the travel up the bore, becomes,

66

Vf =

(wr ) b + u

(ft/sec)

Xf-C-

(ft)

t = - (2.3 log — + e *
a b b

With constrained recoil, assuming a recoil reaction
X we have,

dt

Ubs)

hence

1 Pbd<- Kt = v * pb <'

f -r: — v but / -« vf

therefore,

Kt

Vf - -- = V (ft/sec)

mr

Kt:

2m,

X (ft)

2u

» (2.3 log ^ + _

a b b

+ 2 (sec)

In the several equations, it will be noted, that
the common parameter is the time of travel up the bore
in the gun. Hence if for various values of u, we ob-
tain correspondingly values of time, the free velocity
and displacement is obtained for the given time and
the corresponding effect of the recoil brake during
this time is deducted from the velocity and displace-
ment respectively. Further it has been tacidly as-
sumed that the powder reaction with constrained recoil
is the same as with free recoil at the same time in-
terval. This, however, is not strictly true since the

67

powder reaction is somewhat modified due to the
slightly different motion of the gun with constrained
and free recoil respectively. The effect, however,
is entirely negligible as compared with the magnitude
of the reaction and other factors involved, even with
the most refined measurements and analysis.

EXPANSION OF THE GASES AFTER The manner of
THE SHOT HAS LEFT THE BORE the expansion of the
AND ITS EFFECT ON THE RECOIL powder gases after

the projectile has
left the bore is
very difficult to calculate, and various assumptions

based on empirical data have beeri formulated, for
calculations during this period.

The following theory though imperfect gives an
idea as to the manner of the expansion of the powder
gases in the "After effect Period".

(1) The momentum imparted to the gun

during this period evidently equals the
momentum iwparted to the powder gases:

n>r<Vf * Vfo> ' S<vw - I )

where Vf = maximum free velocity of the recoiling

parts. (ft/sec)
Vf0 = free velocity of recoil when the shot

leaves the bore (ft/sec)
vw = mean velocity of the powder gases attained

(ft/sec)

i = mass of powder charge (Ibs)
v = muzzle velocity of projectile (ft/sec)
Since

mr Vj0 =(n + - )v we have mrVf = mv + m vw

In other words, the maximum free momentum obtained by

68

the gun, equals the sum of the total momentum of the
projectile and the total momentum of the powder
charge.

It is important to note that the momentum
relations are very nearly true provided we are able
to calculate vw the mean velocity of the powder gases
and can neglect the small effect of the air pressures
exerted on the gases.

(2) We have the following energy

relations due to the expansion of the

gases :

(a) Initially the gases have a

Kinetic Energy = * (-)v*
2 3

(b) The work of expansion of the
gases in expanding from the pres-
sure in the bore when the shot
leaves the gun (i.e. the muzzle
pressure) to the atmospheric
pressure, becomes

va

We = / pdV
a. fee y,

ro

where Vo = volume of powder
chamber + volume of the bore of
the gun.

Va= volume of gases at atmospheric
pressure .

(c) The final Kinetic Energy of the
gases may be approximately assumed
equal to: 1 Sv^.

2

It is to be noted that the final Kinetic Energy
of the gases is difficult to calculate due to the di-
vergence or cone effect produced when the gases expand
into the atmosphere The total Kinetic Energy equals

69

the sum of the Kinetic Energy of the center of gravity
of the gases plus the relative Kinetic Energy of the
gases relative to the center of gravity.

From a series of experimental tests conducted by
the Navy on the velocity of free recoil with guns of
various caliber it has been ascertained that the
momentum effect of the powder gases is equivalent to
the weight of the charge times, a constant velocity of
4700 ft/sec.

Assuming the divergence of the spreading of the
gases to be similar at all muzzle velocities , it is
possible to estimate the divergence factor and then in
guns of very high muzzle velocities we may calculate
the maximum free velocity by multiplying the work of
expansion by the divergence constant and the solving
for the mean velocity of the gases.

The pressure of the gases rapidly falls to the
atmospheric value or approximately this value, before
the divergence of spread of the gases is appreciable,
hence the maximum Kinetic Energy of the gases will be
attained at approximately atmospheric pressure.

The change in Kinetic Energy of the powder cases
therefore, becomes,

•jSvw---v = change in Kinetic Energy, and

the work done on the gases, equals the work done by
the external pressures po and pa and the work of ex-
pansion pdV. Hence,

vd

p0V0 - Pava * J Pdv ~ totai work done.
vo

To allow for the relative Kinetic Energy due to the
spreading of the gases, we may multiply the work done
on the gases by a constant, and then equate this value
to the changes of the translatory Kinetic Energy
of the guns.

70

v -n v + f a PdV) =-mv2 -im.,2
o vp pa va J 2 w * q v
V

vo

where K = the divergence constant to allow for the
spreading of the gases at the muzzle. Now the work
of expansion, becomes,

,Va Po vo ~ Pa Va
We = / T> d V = :

'o

where the expansion exponent k = 1.3 approx. Hence
the total work done on the gases, becomes,

Po Vo - Pa Va k .

Po vo - Pa v« + ; = iT^I (povo ~Pava)

K *~" X

further, since prt V ^ _ n yk

0 o pa va, we have,

P "

rrr-fc" v° " Pa Va) = rri p° vo[1 - (-^ ) ^ i

PO

Hence the energy expression reduces to the convenient
form,

p ^L_I_1_

K[r— — r po »o\l - ' / k >J=__va
PO 2 3

from which knowing po» VQ, pa m and v enables us to
immediately calculate vw, the mean free velocity of
the powder gases.

To evaluate the dispersion constant, to take
care of the relative Kinetic Energy of the gases
after expansion, the ballistic data of the 155 m/m
Filloux gun has been chosen, since assuming a mean
velocity of the gases 4700 ft/sec., calculated and
experimental results were found to check very close-
ly.

71

Weight of powder charge w = 26 (Ibs)
Volume of powder chamber S = 1334 (cu.in.)
Total length of bore u = 186 (in.)
Muzzle velocity v = 2410 (ft/sec)
Area of bore Aj, = 29.2 (sq.in.)
Weight of projectile = 96.1 (Ibs)
Max. powder pressure pm = 35300 (Ibs/sq.in.)
Mean Powder pressure =

wyg _ _ 19200 (Ibs/sq.in.)

" "e
644 uAv

Twice Abscissa of maximum pressure

.27 "m *i\ j. -i/^ i \ 2 IT = *^7 ^fl

Muzzle pressure when shot leaves muzzle

_27 2 _u 27_ 2 185.68 x 35300

P° = ~4~ S (e + u)3 P7n " T~ 57.38X (57.38 + 185. 68)3

10140 Ibs/sq.in.

we have then,

0.3
K 32.16 [— « 10140 x U4 VIl -

0.3 V10140

= i 2 i 26 2

• - x 26 x x — x

2 4700 2 3 2410

I

Solving, we have,

156 x 1Q« K V0 = (287 - 25) 10 6 = 262 x 1Q«

72

Hence K = = 0.430

- 3.915 cu.

Hence the energy of translation is but 43* of the total
Kinetic Energy of the gases after complete expansion.
Therefore with guns of numeral ballistic relations,
we may estimate the mean translatory velocity of the
gases after complete expansion, by the formula:

o

where v = muzzle velocity (ft. see)

w = weight of powder charge (Ibs)
b = 1.3 approx.
. pa = atmospheric pressure = 2116 (Ibs/sq.ft.)

po = muzzle pressure of powder gases (Ibs/sq.ft.)

VALLIERS The hypothesis of Vallier assumes,

HYPOTHESIS that during the "after effect Period"

in the powder period of the recoil, that
the powder reaction on the gun falls
off proportional to the time. That is,
If Pob = the total breech reaction of the powder gases,

when the projectile leaves the muzzle (Ibs)
to ~ time of travel of the projectile to the muz-
zle (sec)

tt = total powder period (sec)
P^ = powder reaction on "breech (Its)
t = corresponding time (sec)
then

?b = pob - c<t -to) VALLIERS HYPOTHESIS

where p _ - p

rob Pb rob

c =

How the momentum imparted to the recoiling parts by
the gases during the after effect period, becomes,

73

tt

Pfc dt » mr(Vf • _ Vfo)

where Vfi - ^ax> free veiocity of recoil at end of
* powder period.

Vfo = Free velocity of recoil when the shot

leaves the muzzle.

/ l [Pob f _t (t - t0 ))dt = «r(Vf , -VfQ)

1 Q t O

Integrating, we have,

P b( t-*t.)

-i . «r(V£I VQ)

2mr(Vf, -Vfo)
hence t, _ t ,, __L_J - H_ (sec)

and

pob

C =

2(Vf. -Vfo)mr

Therefore the powder reaction during the after effect
period, becomes,

o

Pb = pob — ^ — ; (Ibs)

2(Vf. - Vfo)mr

RECAPITULATION Of PRIKCIPLE FORMULAS OP

INTERIOR BALLISTICS PERTAINING TO

RECOIL DESIGN.

The velocity and displacement of the recoiling
parts during the travel of the projectile up the "bore
have the following relations with the velocity of the

74

projectile up the bore and the relative displacement
of the projectile in the bore.

(Weight in Ibs. )
If m = massof projectilev 00 .,

oc . la

m = mass of powder charge

mr = mass of recoiling parts"

v = velocity of projectile (ft/sec)

u = displacement of projectile in the bore from

its breech position
V = velocity of recoiling parts (ft/sec)

X = free displacement of recoiling parts (ft)

then =: ra
(m + - )v m + ;
V = 2 = £_ v Approx. (ft/sec)

ra m
m,. + •= mr

• , ffl

(m + - )u m + -

X - = u Approx. (ft)

The pressure on the breech, in terms of the pressure

on the base of the projectile, becomes

If

Pb = breech pressure (total) (Ibs)

P = pressure at base of projectile (total) (Ibs)

m
m * -

P., = P = 1.12 P approx. (Ibs)

m

The mean pressure in the bore, becomes,

I

m + -
4

Pm = pb ^ 1 — ) (Ibs)

75

For building up the energy equation, we are concerned
with the various equivalent masses of the moving ele-
ments that the powder reacts on in terms of the major
mass of the projectile.

The equivalent mass of the projectile, becomes,
if k = radius of gyration about its longitudinal
axis (ft)

p = pitch of the rifling "

9 = pitch angle of the rifling

. 4k2tan2 9 , lh_
m = (1 + - )m = (1 + - ) a |i«t
P2 d '

If we include the effect of the friction of the rifling
we have,

sin 9 + u cos 6 4 tan 9 k2 , Ibs .
m " =[1 + ( - 5 - : - ^) - — - ] m ( - )
cos 9 - u sin € d2

The equivalent mass of the powder charge,
for the energy equation = 35

3 <i*I)

for the momentum equation = -

-

The equivalent mass of recoiling parts become,
(m + =2

ffli

t

The differential equation for the motion of the pro-
jectile up the bore becomes in terms of the mean pres
sure in the bore Pm and the relative displacement u,

(m + | )2

. _ 2 m . dv
Pm = [m " *• - +, - ] v -

mw du

76

If W » the potential energy of the gases at any instant,
we have further,

- dW • pm du * JdQr

where fir»* neat lost "by radiation. The energy equation
for the expansion of the gases, becomes,

where w » ( pm du + J ) dOr (ft/seo)(Adiabaticexpansion)

w » weight of gases (l*bs)

cva specific heat for constant volume (B.T. U.

per Ibs. per
degree)

Op* specific heat for constant pressure (8. T. U.
per Ib. per degree)

Ti » Initial temperature (degrees)

Vi * Initial volume (i.e. volume of powder cham-
ber)

o where 0.7854df * area of bore.

0 .

u * displacement of projectile up the bore.

J * mechanical equivalent of heat = ( 778 ft. Ibs.

per B. T. 17.)

R » (Cp - Cv) J » the gas constant (ft. per degree)
The torque reaction of the projectile in travel-
ing up the bore becomes,

77

mv*
1.05 - r (Ibs.ft)

where r « radius of tho bore (ft)

v = muzzle velocity (ft/sec)

g » number of calibers per revolution

LEDUC'S FORMULA Leduc's formula gives results
sufficiently accurate for recoil
design. The formulas derived from
it are compact and sufficiently
short to "be readily used in ordinary
practical design. These formulas have been used in
the development of the various recoil formulas in the
subsequent chapters. ?or recoil or gun design:
let

v = velocity of projectile up bore (ft/sec)
u = travel up bore (ft)

v0= muzzle velocity (ft/sec)

UQ« total travel up bore (ft)
t = time of travel up "bore (sec)
tQ3 time of total travel up bore "
w = weight of powder charge (Its)
w = "of projectile
Vc 'Volume of powder charge (cu. in.)

A = Density of loading

P » powder reaction on base of projectile (Its)
Pb = " reaction on base of breech "
Po}jsPressure on the projectile when the shot

leaves the muzzle (Ibs)
Pm =Maximum powder reaction on projectile (Its)
Pe= Mean powder reaction on projectile "
A(j= area of bore (sq.in.)

Pm= Maximum given powder pressure from

25000 to 33000 Ibs/sq.in. (Ibs/sq.in.)
Given,

pm, VQ, Vc, w, w and UQ

78

To evaluate:- v, P and t, then,

27.68 w
(1) A-

__ (2) a = 6823 (->! A* "

w

(3) Pm = PmAd (Ibs)

2
W V

(4) Pe = — £— » g = 32.16 ft/sec-2

.

(6) v = ^L- (ft/sec)

b+u

w a2bu

(7) P = (ibs)

g (b + u)3

w aabu

(9)

g (b+uQ)3 (Ibs)

vr'iC ;O «**a f-
-

(10) t = - (2.3 log — + £ +2) (sec)
a b b

b 2U0 U0

to = ~ (2.3 log — - + — + 2) (sec)
a b b

(ID t0 « - H®

2 VQ approx.

The equations of velocity, displacement and time
of free recoil during the travel up the bore, becomes,

79

w
w + -

p a*j

Vf = ( ^)( ) (ft/sec)

wf b + u

» *5

Xf = ( -) u fft)

wr

:*jp»-~ee • • • .-••

20 u

, = 1 (2.3 lo? r- + 5- + 2) (sec)
a

With constrained recoil, assuming a recoil reaction K

Kt

V = V* - — (ft/sec)

X = Xf - — (ft)

2mr

t = (2.3 log + £ * 2) (sec)
a bo

Theexpansion of the gases after the projectile
leaves the hore causes an additional recoil effect.
The hypothesis of Tallier assumes the powder reaction
to fall off proportionally with the time. On this
assumption:

If

Vf ~ the velocity of free recoil at the end of
the powder period.

Vfo = the velocity of free recoil when the shot

leaves the muzzle.

p

tt and t0 the corresponding terms ---

(sec)
ob

.' " - P'01>

2(Vfl _ V)Br (IT,.)

80

» >y

THE PARABOLIC TRAJECTORY The nucleus of exterior

ballistics is the differential
equations of the parabolic
path of a shot projected
in a vacuum. These equations

then nay be modified for air resistance and gyroscopic

deflections due to the angular momentum of the pro-

jectile and air reaction:

Let x and y be the horizontal and vertical coordinates

of the trajectory.

n * the mass of the projectile.

Vo = the muzzle velocity.

t » the time of flight.

0' = the angle of elevation from the horizontal

of the axis of the bore.
0 = angle of elevation of the departure of the

projectile from the muzzle.
e * the increment angle or "jump" to the elastic

deformation of the carriage and the move-

ment of the gun in a direction not along the

axis of the bore. 01- 0
r = angle of sight.
Oaa line of sight.
L * range to given target.
LQ = horizontal range corresponding.
• = striking angle from horizontal.
m1 = angle of fall from line of sight.

The differential equations of motion give:
d»x d«y

Integrating successively, we have,

~ » V0 cos 0 ~ = - gt + V0 sin 0
dt

and

. \\

82

^^^^^

x = V0 cos 0 t y = - - + VQ sin 0 t

<Q

Hence g x2

y = - - + x tan 0 (1)

2 V2Q cos2 0

The general parabolic equation of the trajectory
in vacuum. For maximum range, x being a function of
0, we have

— = 0, when y = 0 in (1)
d0

that is, 2V2Q V2

x = sin"0 cos 0 = — °- sin 20

g g

and

dx 2V*Q

— - = cos 20 = 0

d0 g

Hence, cos 2 0 = 0 or 0 = 45°

When air resistance is considered maximum range for
ordinary guns is obtained at angles which may be from
about 42 to 55 degrees. If x and y are the coordinates
of some target point, we have for the equation of the
line of sight, that

y = x tan r
Substituting in (1)

X2

« COS20 +

gx

7""* .., = tan 0 - tan r
2V_ cos*0

and ya » xa tan r and L = - xa2 - y

which gives the coordinates and ranges in terms of

the muzzle velocity, angle of departure and angle of sight,

CHAPTER III

EXTERNAL REACTIONS ON A CARRIAGE DURING RECOIL
AND COUNTER RECOIL -
STABILITY -
JUMP.

EXTERNAL REACTION The external reaction during re-

coil may "be divided into two primary
periods; that during which the force
of powder pressure on the recoiling
parts exceeds the restraining force

or accelerating period and the retardation period. Again
the period of powder pressure may be divided into the
period of the shot traveling up the "bore to the muzzle
and the after effect period of the powder gases ex-
panding to atmospheric pressure.

Considering the gun, recoiling masses and carriage
as one sustem, the external forces are:

(1) The pressure of the powder gases
along the axis of the bore » ?

(2) The torque reaction due to rifling * T

(3) The weight of the recoiling parts » Wr

(4) The weight of the stationary parts* W&

(5) The balancing reactions exerted by
the ground or platform on the carriage
mount .

If we sum these forces up into X and Y components
and let Hr equal the mass of the recoiling parts, we
have, noting the mass x acceleration of the stationary
parts of the system is nil, ithat)

«,

dt*

83

34

If further, we assume our coordinates along and
normal to the axis of recoil, we have

X = M,

daxr

dt2

Y =0
Equation (1) may be written:

(I1)

(2')

ZX - Mr

d2x,
dt;

= 0

Hence, by the use of D'Alemberts ' principle regarding
the inertia effect, that is, mass x acceleration re-
versed, as an equilibrating force, we reduce the
forces to a system of forces in equilibrium.

Thus by including the inertia effect of the re-
coiling parts as an additional external force, the
problem is reduced to one of statics.

This greatly simplifies the procedure of ac-
curately and quickly obtaining certain overall effects

in stability and the principal reactions throughout a
carriage.
i ;•: irii«9 tea if*t«fl $riiiaa«? ,Jejt. sd: - ,ioO

EXTERNAL EFFECTS Considering now the external
DURING RECOIL reactions upon the total system,
(gun, recoiling parts, and car-
riage proper) including the inertia
of the recoiling -masses, we have
the given forces as shown in figure (1), where

Fig. 1

85

r

P = Total Powder Pressure along axis of bore at

any instant of Powder Pressure Period.
Wr = Weight of recoiling masses "Mr" .
Wa = Weight of carriage proper (includes
stationary part of tipping parts )

H £*_=Inertia force of recoiling masses
r d t

Ha and Va = Horizontal and Vertical components

or equivalent float reaction

7^ = Front Pintle reaction - Horizontal component
assumed zero in order that the reaction may
be determinate.

B = Braking force, resultant hydraulic and re-
cuperator reaction including stuffing box
frictions.

R = Guide frictions = Ri + R2 in diagram.
Ka = Total resistance to recoil for recoiling

masses equals B + R - Wr sin J0 at any instant
during powder pressure period.
Kr = Total resistance to recoil during any in-

stant after P = 0.

During the powder press.ure period, we have for
moments about A, see figure (1)

P(d + e ) - (Hr 1 )d - Wr Lr - Wa La + VbL - 0

d2x
- (Mr 7~7>id * Pe ~ wrLr ~ ffaLa + vbL = °

hence

(3)

In like manner we have after the powder pres-
sure ceases

(4)

Now considering the external reactions on the
recoiling parts alone*, during the powder pressure
period, we have figure (2)

86

Fig. 3.

hence

P -

and

when P » 0 figure (3)

d'x

k dt
d'x

— = * B + R - If- sin j
dt

(5)

,t

B + R - Wr sin 0 = Mr ~

r
dt

(51)

87

Substituting (5) and (51) in (3) and (4) respectively,
we have

K»d + Pe - WrLr - WaLa + VfcL - 0 (6)

Krd - VrLr- WaLa + VbL » 0 (7)

Thus the external effect during the powder pressure
period is always at every instant equal to the total
resistance to recoil, that is, the sum of the total
braking and guide friction, minus the weight component
and a powder pressure couple Fe dependent upon the
actual total powder force.

In general, e is very small and usually for a
first approximation the powder pressure couple can "be
neglected.

Further, for constant resistance to recoil

Kr - Ka - K - B + R - VTr sin 0 (8)

which is the average external effect during recoil on
the total system.

As shown in Chapter VI on the "Dynamics of Re-
coil" i

K « -* — E £ —

b - E + VfT (9)

where Vf * the maximum free velocity of the recoiling
parts, that is

_ W4700 + WQ

wr (10)

W * Weight of powder charge, W * Weight of shot
and Wr • Weight of recoiling parts
v0 * Muzzle velocity of shot
u » Travel up the bore in inches
b * Length of recoil in feet

d a Diameter of bore in inches

E » Unconstrained displacement of recoiling parts
during powder pressure period.

88

T = Total time of powder pressure period.

In equation (9) note that E = Kt Vf T and

K

o

2 v

Substituting these values in (9) and solving for
a wide range of artillery material and thus evaluating
the variables as a function of the diameter of bore,
muzzle velocity and travel up bore, Mr. C. Bethel has
given the very valuable and serviceable formulae, and
accurate to one percent.

Mr Vf 1

= — = - - -

b + (.096+. 0003 d) uvf

vo >*|»H

This formula holds only for constant resistance
to recoil.

It is important to note that the "total braking"
sometimes called "the total pull" is not in general
equal to the resistance to recoil, but is the total
resistance to recoil plus the weight component, that
is

B+R=2Pa+ZPn+2Rs+2fig = K+Wr sin 0 (12)

where 2Pa = Total recuperator reaction
ZPj, = " hydraulic reaction
2Rs = " stuffing box friction

2R,< = Guide friction

K = T M. V,

ir vf

(b - E + Vf T)

To obtain the external reactions on the carriage
mount, it is convenient to know d in the previous
moment formulae about A, in terms of the height of
the trunnions and the distance between the trunnions
and a line through the center of gravity of the re-
coiling parts arjd parallel to the axis of the bore.

89

Let H+ = height of trunnions above the ground

distance from trunnion axis to line through

center of gravity of recoiling parts and

parallel to l>ore.

moment arm of K about A nor.

horizontal distance bet-ween reactions A and B.

from A to center line of
trunnions.
As the gun elevates, we have two cases:

(1) When the line of action K passes
above A, see figure (4)

(2) When the line of action K passes be-
low A, see fig. (5)

t

s =

d =
1 =
c =

-*- K

Fig. 4

90

r

Fig. 5

Fig. 5'

91

For case (1), we note that

h' - (d sin 0 + c ) tan 0 = d cos 0

but s

*' = Ht + ^I~0
and a

E + _f d sln 0 _ c tan 0 = d cos 0

cos 0 cos 0

ht cos 0 + s - d sin2 0- c sin 0 = d cos 0
hence

d = ht cos 0 + s - c sin 0 (13)
For case (2), we note that

h1 + d cos 0 = (c - d sin 0) tan 0
but

n' = ht * — — ^
1 cos 0

cos 0 cos 0

2 2

ht cos 0 + s + d cos 0 = c sin 0 - d sin 0
hence d = c sin 0 - ht cos 0 - s (14)

If W = weight of the total system (gun, recoiling
parts and carriage ), we have for moments about A

*s ^s = vr Lr + wa La In battery

or IB terms of the tipping parts = Wt

and the top carriage alone (not including the stationary

parts of the tipping parts = Wa

Ws Ls = Wt Lt + Va La In battery

where Ls = distance to center of gravity of' total sys-
tem in battery from A

If b = length of recoil, and 0 the angle of
elevation., and Lg - distance to center of gravity of
system out of battery, we have

Ws 14 = Wr (Lr - "b cos 0)+ Wa La

= ¥r Ly + Wa La - ¥r b cos 0
hence TTS L^ = Ws Ls - ^r b cos 0 Out of battery

92

Hence the external reactions at A and B on the
carriage mount become in terns of the resistance to

recoil, powder pressure, height of trunnions and
distance between trunnions and line through center of
gravity of recoiling parts parallel to axis of bore,

For low angles of elevation,
Taking moments about A, we have,
Vb L + Kd + Pe - Ws Ls + tfr(x cos 0)= 0
hence Wg Ls - Wr(x cos 0)- Kd -Pe

Pe disappearing for a finite value of x or in
other words, when pe is used Wr x cos 0 may be
neglected. And since Va r Ws + K sin 0 - V^ or
directly from moments about B, noting that moment
arm of K becomes d'= d + L sin 0 = ht cos 0 + (L-c)sin £l

* S
we have,

Wa(L-Ls) + WP x cos 0 + K(d+L sin 0) - Pe

V 3

Va L

and as before Pe disappearing unless x is very small.
Obviously Ha = K cos 0 and is in no way directly ef-
fected by the powder force.

For high angles of elevation, the moment arm
Kd reverses, and d and d1 become respectively,

d * c sin 0 - ht cos 0 - S
and

d1 » L sin 0 _ d See(fig.5)
= (L-c) sin 0 + bt cos 0 + S
Now taking moments about A and B respectively
Wg Ls - Wr x cos IS + Kd - Pe

b L
and

W8(L-Lg) + Wr x cos 0 + K(L sin 0 - d) * Pe

93

and Ha - K cos 0.

For design use, the external reaction formulae
be conveniently grouped.
IN BATTERY: for low angles of elevation:

j r Nvn* cos 0 + S - c sin 0) - Pe )

•» g Ul a ™ v

) V " "T~

) WS(L-LS) + K(htcos 0 +(L-c) sin 0 +S)+Pe (

) Ha » K cos 0 (

for high angles of elevation:
WgLg+ K(c sin 0 - ht cos 0 - S ) - Pe

( L )

> Ws(L-Lg)+K[(L-c) sin 0 +ht cos 0+S] +Pe (

( Va» ) (16)

) Ha a K cos 0 (

OUT OF BATTERY: for low angles of elevation:
( ya^s ~ wr k cos ^~ ^^ht cos 0 +S- c sin 0 ) )

) WS(L-LS) + Wr b cos 0+K(ht cos j0+(L-c)sin 0+S)

) v ~ T""(17)
c )

) Ha = K cos 0 (

94

for high angles of elevation:
b cos 0 + K(c sin 0- ht cos 0- s

WS(L-LS)+ Wrb COS0 + K(htcos 0* (L-c )sin#+S

L

( Ha = K cos 0 )

These formulae are immediately applicable to
platform mounts traversing about a pintle bearing as
well as field carriages.

In platform mounts, the horizontal reaction of
the platform on the mount is usually taken at the
pintle bearing which is usually located in the front
or muzzle end of the mount. Hence in place of Ha we
have HO = K cos 0. The reactions V^ and Va remain
the same, Va now being the reaction of the platform
on the traversing rollers of the mount. Very often
V)j is divided into two equal vertical oomponents at
the two ends of the traversing arc of the mount, and
in such a case L is the horizontal distance in the
projection of a vertical plane containing the axis
of the bore from the pintle reaction to the traversing
reaction, that is, if L is the actual distance from
the pintle to the other end of the traversing arc,
and 9 is the spread of the arc, then

6

L = L' cos -
2

In a field carriage, for a first approximation
we may assume the horizontal and vertical reaction to
be at the contact of spade and ground. These reactions
are obviously Ha and Va of the previous formulae and
Vfc is the vertical reaction of the ground on the
wheels, and L the distance from the wheel contact to
the spade contact with the ground. For split trails,

95

Va and Ha are obviously equally divided and if the gun
is traversed, a horizontal reaction normal to the
plane of Ha an^ Va is introduced; however, this re-
action will not be considered until later, that is,
the gun will be assumed at zero traverse.

.A closer approximation to actual conditions in
a field carriage is to regard H as acting at a vertical
distance g from the ground line, usually when from 1/2
to 2/3 the vertical depth of the spade in the ground.

The equations then will have an additional moment:

Hag = K cos 0 g,

which is substracted from the moments of the numerator
in the expression for V^ and added to the mpments in
the expression for Va- The general equations for
field carriages are then,

for low angles of elevation:

WSLS- Wr x cos 0 - K.(d + g cos 0)-Pe

vb = .

W3(L-LS) + Hr x cos 0 + K(L sin 0 - d + g cos#)+pe

Va =

Ha = K cos 0

d - ht cos 0 + 3 - c sin 0

and for high angles of elevation:

Y^L, _ Wr x cos 0 + K(d-g cos 0) - Pe

_. *S . ^ _____ _______ ___

L
WS(L-LS) + Wrx cos 0 + K(L sin0-d+g cos0)+ Pe

Ha = K cos 0

d = c sin0 - ht cos0 - S
where Pe disappears if Wr x cos 0 is used or vice versa.

96

Another class of mounts in which the previous

formulae are not applicable, are known as pedestal
or pivot mounts used on Barbette Coast mountings and
for naval guns, as well. These mounts are attached
to the foundation by bolts on a circular base usualljr
equally spaced around the circumference.

With such mounts the question of stability is of
no consideration. The reaction between the foundation
and mount and the distribution of the tension in the
bolts, may be obtained approximately by considering the
base of the mount as absolutely rigid. Then on firing,
the front bolts become the most extended, the deflect-
ions and corresponding stress being proportional to the
distances measured from the back end of the base along
the trace of the intersection of vertical plane, con-
taining the axis of the bore with a horizontal plane,
to the perpendicular chord connecting any two front bolts

Thus if L0, L, etc. are the lengths from the base
end to the perpendicular chord connecting a set of two
bolts, and if jo, j, etc. are the deflections of the
bolts, we have j0: jt: jt: » L0: Lt: L,

Now if the bolts are of equal strength, the ten-
sions are proportional to the deflections, that is

TO : TI: T f * jo: jt: j8 = L0: Lt: L8— ~
that is TQ = C LQ, Tt* C Lt, Q C T C :

Hence the moment about the back end holding the
pedestal down, becomes,

C ll + 2 C L2 + 2 C L* + C Lj = SM

Considering now the gun and mount together we have,

K d - WgLs - Wr x cos 0 = £M

hence Kd -(WSLS- Wr x cos 0.)
C =

L* + 2L2 + 2L2 L2

Oil n

and the maximum tensiqn to which the bolt at the
farther end is subjected, becomes,

97

[K d - (WSLS- Wr x cos 0)]L0

L* + 2 ij + 21; — L;

If the gun traverses 360° every bolt should be designed
for the maximum tension, T0.

The same method may be applied to various other
combinations for holding a gun down on its foundation.

BENDING Itf THE TRAIL In considering the strength
AND CARRIAGE of a carriage body, the reactions

at the trail, Va and Ha, subject
the total carriage to a bending
stress.

This is of special con-
sideration in field carriages of the trail type. The
reaction Va causes bending while Ha decreases the bend-
ing. Hence for maximum bending we should examine the
conditions for maximum Va and minimum Ha.
Now,

WS(L-LS) + K[(L-c)sin 0 + h^ccs 0 + si +Pe

•\f s -^— — ^_^__________^____^________________

a T :

Ha = K cos 0
where

L « horizontal distance between wheel contact and

spade contact with ground (in)
c = horizontal distance from spade to vertical

plane through trunnions (in)
ht = height of trunnions from ground, (in)
s = distance from trunnion to line parallel to

axis of bore and through center of gravity

of recoiling parts (in)
Lg = horizontal distance to center of gravity of

total system, recoiling parts in battery (in)
PS » powder pressure couple (in/lbs)
K = total resistance to recoil (Ibs)
With a field carriage, since the trunnion position
is very close to the wheel contact with the ground,

98

(L-c)sin 0 is always very small compared with ht cos 0,
hence, we have approx.

Ws(l-ls) + K(ht cos 0 + s) + Pe

Va =

If Lx = distance from trial contact with ground to
any section in the carriage body or trail

hv = the height of the section from the ground
y

we have, for the bending moment at section xy,
"xy = Va^x - H hy

Substituting the value for Va and neglecting s being
small, we have L

Mxy = [WS(L-L^) + K'ht cos 0 + Pe^"r~ ~ K cos 0 hy

= WSLX(1- ~) + K cos 0 (JL ht - hy) + Ps^i
BENDINd \N TRWL 8r

Fig. 6

Now from fig. (6) it is evident y- ht is always
greater than hy, hence for maximum bending moment
we must have cos 0=1, that is 0 = 0. Hence the
maximum bending moment occurs at horizontal elevation,

99

At horizontal elevation, ht cos 0 + s = h
henca W-(L-U) + K h + P.

V, =

a L
but we also have critical stability at horizontal
elevation, that is K h + Pe - WSLS = 0 (approx.)
therefore, Va =- Ws (approx.)

that is in virtue of the mount being just stable at
horizontal elevation, or in practice approximately
so, the vertical reaction at the spade equals the
weight of the entire system, gun and carriage to-
gether, yy L _ p

Further Ha = K = S — (ibs)

and the bending moment at section xy in the trail,
becomes, h

lui =WT ^ W T - P ^ * - ( -j n 1 K c ^

neglecting Pe as usually small compared with
W$LS, ws have,

Mxv = W_(L, -L. — ) (in Ibs)

A jr o A. K

For the maximum bending moment in the trail,
we consider the section at the attachment of the
trail to the carriage, then,

Lx = Ls approx. and therefore, the maximum
B. M. becomes, h - h

Li — Iff f f .^^^^^^^^^^^^ l ill

ro-v \T ~ ** c He* \ ) \-Li

a most useful formula in a prelinary carriage layout

It is important to note that if the recoil
varies the above formula and analysis do not hold.
When, however, the recoil varies on elevation the
maximum bending moment in the trail is obtained at
the minimum elevation where the short recoil COTD-
mences, that is, when cos 0 is a maximum for the
minimum recoil.

If b_ = the short recoil at maximum elevation,

then,
we have,

KS - maximum total resistance to recoil, then,

100

W.L^Cl - ~) * K cos 0 (^1 ht - hy) + Pe JL

where Lx = distance from trail contact with ground
to any distance in the carriage body or trail.

h> * the height of the section from the ground.

Pe 3 maximum powder pressure couple.

EXTERNAL REACTIONS DOSING Counter Recoil may
COUNTER RECOIL be divided into two

periods, the accelerating
and the retardation period
so far as the external
effects on the mount are concerned.

During the accelerating period, the external re-
actions on the recoiling parts alone, are the elastic
reaction of the recuperator in the direction of motion,
the guide and stuffing box frictions and a hydraulic
resistance during the whole or part of the accelerat-
ing period, together with the component of the weight
of the recoiling parts parallel to the guides, oppos-
ing the notion of counter recoil.
Hence, if

x = the displacement from beginning of counter
recoil of the recoiling parts with respect

bo guides.
Ffa= the resultant accelerating force of counter

recoil.
Kra the resultant retarding force of counter re-

coil.
Fx * the recuperator reaction for displacement

x trom beginning of counter recoil.
R = the total friction.
Hx3 the hydraulic resistance, if any, of throttling

through recoil orifices or counter recoil buffer.
Then, during the accelerating period

101

d x i i

m j^j- = Fx - R - Hx - Wr sin 0 = Ka

and for the subsequent retardation
. t

- m = R + Hx + W_ sin 0 - F = K'

dt*

Considering now the external forces on the total
system (recoiling parts together with mount) the
braking resistance for the recoiling parts then be-
come internal reactions, and considering inertia as
an equilibrating force, we have, as before the fol-
lowing external forces,

• d*x

Ka = m- — - The inertia resistance during ac-
celeration which is opposite to
C'recoil.

• » d'x

*r md~T* ^ne inertia resistance during

retardation which is in the direction

of C'recoil.

Wr * Wt. of recoiling parts
Wa * Wt. of carriage proper

Ha and Va Horizontal and vertical reactions of
spade and float

y^ * Front Pintle reaction - horizontal com-
ponent assumed zero as before.
During the accelerating period, obviously,
Ka < Kr that is,

FX-R-HX - Wrsin 0 < Fx+ R + Hx - Wrsin 0

hence, so far as stability and the balancing re-
actions exerted i>y the ground or platforu on the
carriage mount are concerned, the external effect
during the acceleration period of counter recoil

102

need not be considered.

If now, the inertia resistance is considered as
an equilibrating force, we have

Kr (d+L sin 0)-Wr [ (L-Lr)+b-x cos 0]- Va(L-La)+VaL= 0

Let d = d + L sin 0 = ht cos 0 + (L- c) sin 0 + S

Hence the limitation for counter recoil stability,
noting that Wr(L-Lr) + Wa(L-La) = Ws(L-La) becomes

Krd' = Ws (L-LS; + Wr(b-x) cos 0

For a constant marginal counter recoil stability

.
moment O1 this equation becomes

Krd' =[G'+WS(L-LS) + Wr b cos 0] - Wr cos 0 x
and the stability slope for a constant marginal counter
.ecoil stability is evidently

, Wr cos 0

that is decreasing as the recoiling masses move into
battery. Minimum stability is evidently in battery
position and 0=0, that is

Wf r r \
s \L> -L>s I

h

where h = d for 0=0

In ordinary field- carriages, the weight of the
system in battery is very close to the wheel axle
or contact of ground and wheel, consequently (L-LS)
is very small .

Therefore counter recoil stability is the
primary limitation in the design of a counter recoil
system.

STABILITY The question of stability

for field carriages is of fundament-
al importance, it being a primary
limitation imposed on the design
of a recoil system. If a gun

carriage is to be stable, then

Kd - WaLa - *r(lr - x cos #) = 0

103

If we have a constant marginal moment G, that
is an excess stability, we have

Kd - WaLa - Wr (lr - x cos 0) = G
K =

-G + VTSLS - x v»r cos 0

= A - m x
A =

where — G + W I W cos 0

\* • » « L » !•«» w W Jv

Thus the resistance to recoil to conform
with a constant margin of stability decreases in the
recoil proportionally to the distance recoiled from
battery.

In battery, the resistance to recoil,
-G * WSLS

Kb = A = —

and put of battery, the resistance to recoil becomes,
where b = total length of recoil

-G + WSLS Wr b cos 0

d d

consequently, fot a constant margin of stability,
Wr b cos 0

Kb

d

From this we obtain the equations of resistance to
recoil for constant stability against displacement,

Wr cos IS -G + WsLg

Kx = Kb x, where Gb =

d

In our Ordnance Department, KX = KQ = a constant

during the powder pressure period.

Thus if B represents the corresponding length of
recoil, then for a constant stability moment G,
-G + WSLS E Wr cos 0

K /-\ "~ v ~ , ' A ~~ 1

104

*r cos 0
and Kx - K0 (x - E)

Obviously the stability "slope" or space rate
of change of resistance to recoil for constant margin
of stability, is ^ cos 0

ID = — — — —

d
where

d = ht cos 0 + s - c sin 0

A3 the gun elevates, Wr cos 0 remains finite,
while d decreases to zero at the elevation Si, where
the line of action of the resistance to recoil passes
through the spade point.

Thus the stability slope "m" thereby increases
to an infinite value at that same elevation.

But it is important to note that the resistance
to recoil out of battery is finite and increases con-
siderably as "d" decreases so far as it is limited
by stability.

Obviously in design it is inconsistent to
•ake the slope of the space rate of change of resist-
ance to recoil consistent with the stability slope
as the gun elevates, since the stability becomes
sufficiently increased to allow a large resistance
to recoil bo be used.

We may, therefore, cause the slope to vary arbitrar-
ily as a linear function from a maximum value at an
arbitrary low angle of elevation, say some value from
0° to 6°, to zero at the angle of elevation where the
resistance to recoil passes through the spade.

Thus if,

0° = the initial angle or lower angle of elevation
from which the slope is to decrease
arbitrarily.

£Jt * the angle of elevation corresponding to

where the resistance to recoil passes through
the spade.
do = moment arm of resistance to recoil about

spade point for angle to

105

8 SOO 0 .j* 8^8 * *J " Z g*

L, tf

m = stability slope for any angle of elevation 0

Wr cos j0

m = - = stability slope at lower angle
o do

of elevation.

m = m0 - k (0 - 00)

then,

m = m0 •

At angle, of elevation 0t, o = 0 hence

•o * k^t - *o> °r k = TO°.

0-0Q

hence

m = mo - (-zr\ TT— ) (# - 00) or substituting for

i " o

Wr cos 00 0 - 00 Wr cos 00 0t- 0

Thus the variation of the space rate of change
of resistance to recoil may be divided into two

periods,

(1) from 0° to 0Q

w •• ~ C^™ ci

Wr cos 0
OB = which is parallel

to the stability slope

(2) from 00 to 0°

Wr cos 08 0-0

,

where the slope

ln W ~ & f\

is arbitrary.

A graph of the variation of the space rate of

change of the resistance to recoil against elevation
conforming' to the assumption (1) and (2).

If there is always to be an excess stability
couple G we have from the previous discussion, fixed
limitations for the resistance to recoil in and out
of battery.

Thus, from 0° to 0°

o

106

- G + WSLS - G + WSL3 Wr b cos ft

Kv = - : k = : -

where throughout recoil G is a constant marginal
stability couple, and from 0O to 0t -
-G + WSLS »r b cos 0

the length of re-
coil being as be-
fore shortened as

the gun elevates but if the stability marginal move-
ment is never to be decreased for any part of the
recoil below G, since the stability slope and space
rate of resistance to recoil increase and decrease
respectively as 0 increases from 0Q to 0 it is ob-
vious that the minimum stability is in the position
of out of battery.

Therefore the resistance to recoil in battery
is the resistance to recoil out of battery with a
marginal moment G of actual stability, augmented by
m b .

That is,

K =

-G + WSLS

Wrb cos 0

Wrb cos 00 ji
+ r

J - 0
1 ]

d

d

do <

»J=*0

cos 0 cos

00 0- 0

f * 1 \

LENGTH OF RECOIL Obviously the overturning
CONSISTENT WITH force, that is the resistance to
STABILITY OF MOUNT recoil, is a function of the

length of recoil varying roughly
inversely as the length of re- v

coil. Hence as the gun elevates the stability in-
creases and the recoil may therefore be shortened.

In a preliminary design it is desirable to know
the length of recoil as limited by stability, from
0° or the lowest elevation wherein stability is de-
sired to the elevation 0° where the stability slope

107

is made to change arbitrarily.

Let Cs = the constant of stability =

Overturning moment

= where the overturn-
Stabilizing moment

ing moment = Krd and the stabilizing
moment = WCL_ - W_ b cos 0

OO I

We may consider the limiting recoil at various elevations

(1) with a constant resistance to recoil

as would occur in certain types of re-
coil systems.

(2) with a variable resistance to recoil
using a stability slope as outlined in the
previous paragraph-

For a constant resistance to recoil : = K :

The critical position of stability is obviously
with the gun at the end of recoil out of battery.

Then CS(WSLS - Wr b cos 0)
K = - -r-

>" -sj • :?f.Q -j^bwcq arfJ $.«iii;fe

1 va

2 mr Vf

K = Ses "DYNAMICS OF RECOIL". Chap. VI.

b-E +VfT

Where E = displacement during powder period in free
recoil.

T = total time of free recoil.

Vf = Max. free velocity of recoil,
hence

CS(WSLS - Wr b cos £5)

b-B+VfT d

The above equation reduces to the quadratic form

Ab2 + Bb + C = 0 and its solution is,

/—*

- B ± / B - 4AC
b = — -

2 A

108

Where A = Wr cos

B = Wr cos 0 (VfT - S) - WSLS
C = WsLs(VfT - E)

For rough estimates, especially where the length
of recoil is comparatively long, we may assume,

^ mrVf CS(WSLS - Wr b cos 0)

- B + /B2 - 4 AC Cs Wr cos

b . » and A . -

C = - mr V£
For Variable Resistance to Recoil:

- - -- -- -----

The resistance to recoil is assumed constant

during the powder pressure period and thence to de-

crease uniformly with a stability slope as given in
previous article. Therefore, from the end of the

powder pressure period to the end of recoil, the
stability factor remains constant from 0 to 0O

(i.e. to where 'the stability slope is made to change
arbitrarily) .

At the end of the powder period. (See Dynamics
of Recoil): a

Kd = G.(»3LS - Wp (E - -—-) cos 0

2m_
hence

CS(WSLS - *rE cos 6)

do I w

-- COS 0

Now the resistance to recoil out of battery at the
end of recoil, becomes,

109

KT*

K - m (b - E + - ) (See Dynamics of Recoil)
2m r

hence by the equation of energy

VT* KT2 KT 9

[2K-m(b-E+ £-)] <b-E+ g-j - Mr(Vf ---- )2
/snip «•• nr

Expanding and simplifying, we have the quadratic
form: Ab* + Bb + C = 0

- B + B2 - 4AC
where b = - — -

H

fr cos 0 0
and A = m = Cs — 7 - from 0 to 0

B = - 2K - 2mB

ID.

c = [2C 2VfT) K +

C(WL - WE cos 0)

From #o to 0t degrees, the stability slope is

made to change arbitrarily, decreasing proportionally
with the elevation from the stability slope at j0o to
zero slope at 0^ where the line of action of the re-
sistance to recoil passes through the spade point.
The critical stability is obviously at the end of
recoil, and the resistance to recoil in battery (K)
is the resistance to recoil out of battery (k) aug-
mented by the product of the length of recoil from
the end of the powder period to the end of recoil
multiplied by the arbitrary stability slope (m).
From the energy equation, we have,

110

2

KT KT

(K + k) (b - E + £L-> = m (Vf - — )
2rar mr

now K = k + m(b-E + — )
2mr

K = 5 — = the constant resistance to

mT
1 _ recoil during the powder

2rar period,

and CS(WSLS - Wr b cos 0)

k = = the resistance to

d

recoil at the end

of recoil.

Substituting these values in the enery equation,
we obtain a quadratic equation in "b!! A sufficient
approximation and simplification can be made, by not-
ing that 2

E - — — - 0.9 E approximately and
2mr

KT
Vf = 0.9 Vf approximately

Therefore, (K +~k) (b- 0.9E) = 0.81 mr vj
and K = k + m(b- 0.9E)

Cs(WsLg - Wr b cos 0)

= + m(b-0.9E)

d

substituting in the energy equation, we have,

2Ca

(W_LS - Wr b cos J0) + m(b- 0.9E)(b- 0.9E) = 0.81 m_Vf

d z

Reducing and simplifying, we have the quadratic sol-
ution, ,—

-B ± / B - 4AC
b = —

2A

s
where A = m - —7- Wr cos 0

2C,

III
2Cf

* °-9E wr cos

2

Q = 0.8l(mE - mrVf

o
from 0 to

Wr cos 00 0t - 0

m , ( ) from 0 to 0

For a close approximation and when the resist-
ance to recoil is not constant during the powder
period, if

K = the resistance to recoil in battery
k = the resistance to recoil out of battery,
we have,

Kl- fe m V ?

r R 111 M V f

( >b = (approximately)

but K = k + mb

cs

and k = 7- 0»_L, - W_b cos 0)

d as

Substituting, we have

20 s
[-•— (WSLS - Wr b cos 0) + mb] b = mrVf

and the value b, becomes,

-B ± /V - 4AC
b =

2A

where pp

3 nr

A = m - ~r~*'r cos «
.fee d

2CS

B = -r WSLS

c = - «rvf8

N

Wr cos 0

ra = " or any arbitrary slope as desired.
d

The above formula is sufficiently exact for a
preliminary layout with a variable recoil and resist-
ance to recoil provided the margin of stability is
chosen fairly large, that is when a low factor of
stability is taken.

JUMP OF A FIELD CARRIAGE When the overturning

moment exceeds the stabiliz-
ing moment, we have unstabil-
ity and an induced angular
rotation about the spade

point. After the recoil period, the gun carriage is
returned to the ground by the moment of the weights
of the system. This phenomena is known as the
jump of the carriage.

For the condition of unstability, we have:

K d ~ *s^s * *r cos > °
where as before,

K = total resistance to recoil

Wg= weight of entire gun carriage including gun
13= distance from spade contact with ground to

center of gravity of total system in the
battery position.

wr= weight of the recoiling parts

x = movement in the recoil of the gun.
To analyse the motion of the system, consider

(a) the recoil or accelerating period.

(b) the retardation or return period.

The recoil period may be subdivided into the

113

powder period and the pure recoil period. During the
recoil period the gun and gun carriage are given an
angular velocity which reaches its maximum at the
end of recoil. During the retardation the angular
velocity is gradually decreased to zero, but with
increased angular displacement, the maximum angular
displacement occuring when the angular velocity
reaches its zero value. Further change in angular
velocity results in a negative velocity and a corres-
ponding angular return of the mount to its initial

position.

ttniliootn jo x-*Y"''^
The acceleration during the recoil period is not

constant, even with constant resistance to recoil,
due to the fact that the moment of inertia and the
moment of the weights of the recoiling parts about
the spade point varies in the relative recoil of the
gun. Therefore, the angular acceleration is not
constant during the accelerating period. Likewise
during the return of the recoiling parts into
battery. Further the effect of the relative counter
recoil modifies the return angular motion.

Consider the reaction and configuration of t"he
recoiling parts and carriage mount respectively.
See figure (7).

114

Let X and Y = the components of the reaction between
the recoiling parts and carriage mount,

parallel and normal to the guides res-
pectively.

M = the couple exerted between same.
Ia= the moment of inertia of carriage mount

about the spade point.

Ir= moment of inertia about the center of gravity
of the recoiling parts.

dx = perpendicular distance from spade point to
* i

line of action of X.

dx = perpendicular distance from X to center of

gravity of recoiling parts.

d = dx + dx = perpendicular distance to line
parallel to guides and through
center of gravity of recoiling parts from
the spade constant with ground.
Q = angle made by d with the vertical
0 = angle of elevation of the gun (in battery)
x = distance recoiled by gun from battery position
x0s distance from "d" to center of gravity of re-
coiling parts in battery measured in
direction of X axis of perpendicular to line
d.
r = distance from spade point to center of gravity

5 J

of recoiling parts,
e = angle r makes with vertical
lr= horizontal distance to center of gravity of

recoiling parts from spade contact with

ground .
Wa = weight of carriage proper (not including

recoiling weights)
ra = distance from spade point to center of gravity

of carriage proper,
o = angle ra makes with horizontal
la = horizontal distance from spade point to

center of gravity of carriage proper.
Then

lr = (xQ-x) cos 6 - d sin 6

115

la = ra cos (9 + a - 0)

where in battery 6 = 0 and for any other angular
position during the jump of the carriage,

6=0+6 B=a variable angle during the jump,
For the angular motion about the spade point,

For the carriage mount, without the recoiling
parts,

d 6
i " "u a~* "** d + 2

Xdx - Y(x0 - x)+ m - wala = Ia T— - (i)

and for the recoiling parts,

adding (1) and (2), we have,

d*6 (3)
Xd - Y(xQ-x)- wala = (Ia + Ip) — -

Since the recoiling parts are constrained to

rotate with the carriage mount, they partake an
angular acceleration about the spade point combined
with a relative acceleration along the guides.

The acceleration of tne recoiling parts is

AfT Jtfc> 9&
divided into:

(1) The tangential acceleration of the
recoiling parts about the spade point;
due to the constraint in the guides,

dt2 a°d is divided into components in the
x and y direction

d*e /« j d2e )

- cos (e + e) = d — -— f
dta dt2

116

(2) The centripetal acceleration of
the recoiling parts about the spade
point due to the constraint in the
guides,

s ,de . 2
rw - r (. )

dt and divided into components in

the x and y direction.

,d9,* , N,d6v!

r(— ) sin (9 + e) = (x0 - x)(— 4

r(- — )* cos (e + e) = d
at

(3) The relative acceleration of the
recoiling parts

d'x <*vr

—— — - -T — along the x axis

(4) The relative complimentary centri-
petal acceleration due to the combined
angular and relative motion of the re-
coiling parts:

de

(5) The angular acceleration of the
recoiling parts which obviously equals
the angular acceleration about the
spade point, that is

d2e

dt!

For the motion of the recoiling parts
along the x axis, we have

dvr d%
Pb - m_ - ard 4 wr sin e - mp(xQ-

dt dt«
0 (4)

117

For the motion of the recoiling parts
normal to the guides,

x d*e de ,,de .a

Y - ra_(xn-x) -- wr cos 9+2 nrvr - + m_d(— ) =
dt» dt d t

0 (5)

Substituting (4-) and (5) in (3) we have,

1 2

2 2

ur 1 — mi 1 + 9m v ( v — v ^

- - aL . _ at_ "r^r Wa1a 60Brv,Axo x;

*9 d29

(Ia+Ir) — - = 0 (6)

dt dt
where

Ir=(x0~x) cose - d sin 0 )

' functions of the

, . variable angle

la =ra cos (e+a-6) ) e

From equations (4) and (6), we have 8 as a
function of t. An exact, solution of these dif-
ferential equations is complicated and therefore an
approximate solution must be resorted to.

APPROXIMATE SOLUTION 09 THE JUMP Of A FIELD
CARRIAGE.

The static equation of recoil, that is the
equation of motion of the recoiling parts upon the
carriage is stationary, becomes,
s

>• W_ sin e -XR * 0

and the equation of motion of the recoiling parts
along the guides when the carriage jumps, becomes,

9 - X - o,rd L£ -«r <x0-x> (A 0

118

Now the term mr(xrt-x)( — ) is small and may be neg-

dt
lected, but on the ,2

d 9
other hand the term m_d may be considerable.

dt2
Furthermore the

braking X and Xs may differ considerably as well.
The term 2

Pb-mr - — -= Ks in static recoil.

whereas with the jump of a carriage

,2

Pb- mr = cKs where c = 0.9 approx.

dta

During the pure recoil on retardation period of
the recoiling parts, we have

m_- — - = K, in static recoil.
rdt2

whereas when the carriage jumps,

,2

m_ — - = cK, where c = 0.7 to 0.9
r dt8

Considering the moment equation for the movement of
the total mount about the spade point, we have,

(pb-rar — * )d-[mr(d2+(x0-x)2)+Ir+Ia] +2mrvr(xo-x)

dt2 dt2

d6

_ **rlr _ „ I _ Q

dt a a

where lr=(xo-x) cos 6 - d sin 6

la=r_ cos (0 + a -v )

de
The term 2mrvr(xo-x) — is always small and may be

Q t

neglected .

If b = length of recoil, for the average during
the recoil,

let

b

v — y — Y — •«-

*o x ° 2

119

Then, we have, for an approximate solution,

2

cKd-[mr(d%(x0- |)2 + Ir + Ia] i-i - wsls = 0 (7)

where wsls = «ala + «rlr c = 0.8 to 0.9

and ws = wa + wr and approximately, if the jump is
small,

lr= (x0- -) cos 0 - d sin#

_ . ~ _ . .. i ' t

la= ra cos a

0 = angle of elevation of the gun

b = length of recoil

d = perpendicular distance of line parallel

to guides and through center of gravity

of recoiling parts from spade contact with

ground.
XQ= the perpendicular distance from d to the

center of gravity of the recoiling parts
ra = distance from spade point to center of

gravity of carriage proper
a = angle ra makes with horizontal
Hence for the angular acceleration,

d29 _ eKd- wsls _ rad

,2 x,2 D72 I I -

dt mr(d +xo- -) +Ir+ Ia sec*

If we assume a constant acceleration, we have,

for the angular velocity attained at the end of recoil,

d0 (cKd - w,,ls)t rad

( \

At 2 1

mr(d(x0- |)-Ir-Ia
where t^ = the time of recoil we have approximately,

cKs

wv -H w 4700
where V = 0.9

120

w= weight of projectile (Ibs)
w= weight of charge (Ibs)
ws= weight of recoiling parts (Ibs)
c s 0.9 approx. and tp= the total powder period ob-

tained by the methods of
interior ballistics.

The angular displacement during the first period
of the jump, becomes,

t (cKd-wsl3)t*

6 = - B radius.

-

During the second period of the jump we have,
the angular velocity decreasing but the angular
displacement still increasing: then

rad

~

Integrating, we have

t «r(d»(x0- |)2)+Ir+Ia
and for the angular displacement,

SI *,!< (- -Ox* L)^m it
- ws!3 t de

- *)2) +1, +1, + ~ fc " 9l

To determine the time of jump required to attain
the maximum angular displacement, we have the
angular velocity reduced to zero, whence,

aet wsis ta

» j i. ' ~ ,

from which we may determine ts> Therefore, the

121

maximum angular displacement, becomes,

_I *rde

mr^d2+(x0- '
2

tf

The effect of counter recoil is to increase t

2

and decrease the negative moment (- w0l ).

S S

RECAPITULATION OF FORMULAS:

EXTERNAL EFFECTS AND STABILITY,

Resistance to recoil:

J Mf

K = (Ibs) constant resistance through-

b-E+Vfr

out recoil.

mp= mass of recoiling parts = wr (16s)

I

g = 32.16 ft/sec. a
b = length of recoil (ft)
E = free recoil displacement during powder period

(ft)

T = time of free recoil (sec)
Vf= Max. velocity of free recoil ft/sec.

BETHEL'S FORMULA

. mrvf 1

(Ibs) Constant

lD + (.096 +. 0003d )M Vf resistance.

Vo

M = travel up bore (inches)
VQ= muzzle velocity (ft/sec)

d = diam. of bore (inches)

Assuming a gun carriage to be supported by a
hinge joint at the rear (A) and a vertical support

122

in the front (B) we have the following equations for

the reactions of the supports:
Let

Ha-= horizontal component at rear hinge support

or spade of carriage. (Ibs)

Va= vertical component at rear hinge support (Ibs)

Vjj= reaction of front support assumed vertical

(Ibs)
L = horizontal distance between carriage supports

(in)
ht= height of trunnion above support (in)

s = perpendicular distance from center of

gravity to recoiling parts to line of action
of the resistance to recoil (in)

c = horizontal distance from rear support to

trunnion (in)

K = total resistance to recoil (Ibs)

<& = angle of elevation of gun

g = vertical distance from ground to horizontal
.component of resultant spade reaction.

IN BATTERY: For low angles of elevation:

w<.L»-K(h+ cos 0 + s-c sin 0)-Pe '

:v - ' -

K[ht+ cos 0+(L-c)sin

) Ha = K cos 0

123

For high angles of elevation

wsL, +K(c sin 0 -ht cos 0 - s)- Pe )

) Vb = (

ws(L-Ls) +K[(L-c) sin 0 + h + cos 0+s]+Pe < (Ibs)

( va - -r ~)

( Ha = K cos 0 )

OUT OF BATTERY: For low angles of elevation:

w"3Ls - Wrb cos 0-K(ht cos 0 + s -c sin 0

( WS(L-LS) +W_b cos 0 +K(b+ cos 0+(L-c)sin0+a>

) Va = - *— - (

) (

( Ha = K cos 0 )

For high angles of elevation:

( )

) wgLs-»rb cos 0+K(c sin 0 -ht cos 0-s) (

( Vb • - L - )

) (

( »?g(L-Ls) +»rb cos 0 +K(htcos 0(L-c)sin0+s) )

Ha » K cos 0 (

With a field carriage where the spade is in-
serted in the ground, the center of pressure lies a
distance "g" inches vertically down. The general
equations for the support of a field carriage,
therefore become,

124

For low angles of elevation:

wsLs- wrx cos 0-K(d+g cos 0)- Fe

( b L )

ws(L-Ls)+wr x cos 0+K(L sin 0-d+g cos0)+Fe
( Va= )•

( Ha = K cos 0 )

( d = ht cos 0 + s - c sin 0 )

For high angles of elevation:

\ - *sLs wr x cos 0 *K(d-g cos 0)-Fe
; V^_

( Va=w(L-Ls)+wr x coS0+K(L sin0 -d+^ cos 0)+Fe )

) (

( Ha= K cos 0; d=c sin 0-htcos 0 - s )

In certain types of Barbette mounts, we have
the bottom carriage held down by tension bolts to a
circular base plate. If we draw a series of
parallel chords through the bolts on either side
of the axis of the gun, and if we let the distance
from those several chords measured from the rear
bolt, be LQ, L - — - - I»n. we have, for the maximum
tension induced in a tension bolt given by the ex-

[Kd-(wsL3-Wr x cos 0)JL0

TO = — — : — :

BEHDINa III THB TRAIL AND CARRIAflB.

Considering the section at the attachment of
the trail to the carriage, for a constant length of

125

recoil the maximum bending in the trail occurs at
horizontal elevation and is given by the following
expression: h_h

B.K. at the attachment
of trail to carriage.

h = the "height of the center of gravity of the
recoiling parts (axis of bore practically above the
ground when the gun is in its horizontal position.

hvs= the height of the neutral axis of the
section above the ground.

ws- weight of entire mount including the gun.

Ls= horizontal distance from the spade to the

center of gravity of the weight of the entire

mount.

When the recoil varies on elevation, the maximum
"bending moment in the trail is obtained at the minimum
elevation where the short recoil commences, we have,

^s kx Lx

Mxv=wsLx(l )+Ks cos 0S( — ht-hv)+Pe — -

L L !-•

where

Ks = maximum total resistance to recoil corres-
ponding to short recoil "bs.

0S= minimum angle of short recoil.

Ls= distance from trail contact with ground to
any distance in the carriage body of trail.

hy= the height of the neutral axis of the sect-
ion from the ground.

Pe = maximum powder pressure couple.

STABILITY OP COUNTER RECOIL.

In the design of a field carriage counter re-
coil stability is a basic limitation. We have for
counter recoil stability that.

126

The equation stability, gives, for variable
resistance to recoil, for low angles of elevation
consistent with the stability slope,

-8+ /B* -4AC
b =

2 A
where

ff_cos 0 o ^o

A = • = Cs -*— (from 0 to 0O elevation)

B» SSI - 2K-2mE
mr

and

CS(WSLS- wr E cos 0 ) (Ibs)

w T2
d - C3 — E- — cos J25

2rar

After an arbitrary elevation 00 (approx.5 ) the
stability of the mount greatly increases with
elevations and therefore the stability slope is made
to arbitrarily decrease with the elevation arriving
at constant resistance to recoil at the elevation
corresponding to where the line of action of the
resistance to recoil passes through the spade point.
To estimate the minimum recoil allowable for the
various angles of elevation in this range, we have

-B± /B2-4AC
b =

2A . jsstaas jjaiJ BC

0 J0.-0

from<.00 to

3 * ^

A = D — - 5f_ COS 0

d

2Ca
B = (waLs* P.9E wr cos 0)-1.8 mE

0.81

127

< • (L-t.)

Rr " "~"h" where Kr= the total resistance

of counter recoil at
horizontal elevation.
ws= weight of entire mount including gun.

L3= horizontal distance from spade to center of

gravity of ws.
L = horizontal distance from spade to wheel

contact with ground.

Further 2

„ d x

(4— f- may be obtained from the velocity
curve of counter recoil towards
the battery position).
and Kr= Hx +R+wr sin 0 - Fx where Hx = hydraulic or

buffer brak-
ing at end of counter recoil. (Ibs)

R = total friction resistance

wr sin 0 =0 weight compound equals zero at

horizontal elevation.
Fx= recuperator reaction. (Ibs)

RECOIL STABILITY

The stability limitation of the resistance to
recoil varies in the recoil due to the movement of
the recoiling weights. The slope or rate of the
variation in the recoil of the equivalent force
applied through the center of gravity of the re-
coiling parts and parallel to the guides that will
just overturn the mount, is given by the following
expression:

wr cos 0
ra= from 0° to 0

128

where

ra = the stability slope
0 3 angle of elevation

d = perpendicular distance from spade to line
through center of gravity recoiling parts
parallel to the guides.
wr=» weight of recoiling parts

00= the initial angle or lower angle of elevation
from which the slope is to decrease
arbitrarily.

If from 0O the slope is made to decrease ar-
bitrarily with the elevation, to the elevation 0,
the angle of elevation corresponding to where the
line through the center of gravity of the recoiling
parts parallel to the guides passes through the spade
point, we have for the stability slope

wr cos 00 0t- 0

m = (— — ~r— ) where the slope is

d 0 ~0

arbitrary.

LBMQTH Of RECOIL COM3ISTBNT WITH STABILITY
OP MOUNTS.

The equation of stability, gives, for constant
resistance to recoil,

-mrVf _Cs(wgLs -wrb cos

The solution of this quadratic equation for b, gives:

-B± /B -4AC

b =

2A

where A= i»r cos 0 ) where all units

8= wr cos 0(VfT-£)-wsLs ( are in feet
..2s ) and pounds.

_ i"r»i ci

)

CHAPTER IV

-soo

INTERNAL REACTIONS.

<: - ' '

In the design of the various parts of a gun
carriage it is of fundamental importance that we
have a coraplste knowledge of the stresses to which
each member is subjected, and the variations of such
throughout recoil and the position of elevation and
traverse .

We have already considered the external reactions
on the whole system, and such reactions are useful in
computing the stresses in the supporting structure
for a gun mount as the strength of concrete emplace-
ments for barbette mounts, or the strength of a rail-
way car or caterpillar frame.

The primary internal reactions within a gun and
its mount may be classified as follows:

(a) The mutual reactions between the recoil-
ing parts and the carriage proper or gun
mount.

(b ) The mutual reaction, between the tipping
parts or cradle and the top carriage.

(c) The mutual reaction between the top
carriage and bottom carriage.

The mutual reaction (a) is between the moveable
and statipnary part of th3 total system during" the
recoil; that of (b) between the moveable and station-

ary parts during elevation of the gun; and that of
(c) between the moveable and stationary parts in
traversing the gun.

The mutual reaction (a) may be subdivided into
individual or component reactions as follows:

(.1) The reactions of the constraints
due to the guides or clip reactions at

129

130

the two ends of the clips in contact
with the guides, which may be subdiv-
ided into friction and normal com-
ponents .

(2) The mutual reaction of the elastic
medium connecting the recoiling parts
to the carriage proper, that is, the
hydraulic brake and recuperator re-
action, together with the joint
frictions. This will be known as the
elastic reaction between the recoil-
ing parts and carriage proper.
The mutual reaction (b) may be subdivided into:

(1) The trunnion reaction between the

tipping parts and top carriage.

(2) The elevating arc reaction between
the elevating arc of the tipping parts,
and the pinion of the top carriage.

The mutual reaction (c) may be subdivided into:

,(1) The pintle or pivot reaction between
the pintle bearing on the bottom car-
riage or platform mount and the pintle
of the top carriage fitting within this
bearing.
(2) The traversing arc reaction, that

is, the reaction between the traversing
arc of the top and bottom carriage.
These are usually roller* reactions for
platform or pedestal mounts, the rol-
lers being either a part of the top or
bottom carriage or else clip reactions

field carriage and may be more
or less distributed about the arc of
contact.

Let X and Y - the coordinates of the center of
gravity of the recoiling parts along end perpendicular
to the guides with origin at center of gravity of re-
coiling parts.

131

xt and yt = coordinates of front clip reaction
measured from the center of gravity
of the recoiling parts.
&t = Normal component to guides of front clip

reaction.
uQt = Frictional tangentional component of front

clip reaction.
x and y = coordinates of rear clip reactioa

2 2 ^^^

measured from the center of gravity
of the recoiling parts.
Q = Rear clip reaction normal component.

u&2= Rear clip reaction frictional component.
B = nQ + nQ = total guide friction.

1 2

B = elastic reaction (hydraulic breaking and re-
cuperator reaction including friction of
joints) assumed parallel to the guides.

F = the total powder pressure on the breech of

the gun which necessarily lies along the axis
of the bore.

e = the perpendicular distance from center of

gravity of recoiling parts to line of action
F, that is to axis of bore.

Assuming as in Chapter III, the mount to be

hinged at the rear or breech end to its support and
resting on a smooth surface at the front end, and if
d = perpendicular distance from hinge to line

through center of gravity of recoiling parts,
parallel to guides.
djj= perpendicular distance from hinge to line of

action of B.
lr= horizontal distance from hinge to center of

gravity of recoiling parts in battery.
la= horizontal distance from hinge to center of

gravity of stationary parts of system (includes
stationary parts of tipping parts).
From fig.(l) considering the reactions on the re-
coiling parts alone, we have from the equations of
motion:

132

Fig. 1

133

for notion along the x axis, 2

.F"— B — uQ - uQ + w*r sin 0 = mr— — (1)

and since there is no motion along the .y axis,
.-••; -ie* adi oj ; .- 'r^ i»fljft?j';^ i'-'ic'~

Qf- Qi = wr cos ^ (2)

tdJ *c

and taking movements about the center of gravity since
there is no angular acceleration

B(d-dh)-u(Q y -Q y )+Fe-Q x - Q x =0 (3)

u 22 1122

Now in fig.(l) considering the gun carriage or
mount including the stationary parts of the tipping
parts, we have for the moments of the reaction of the
recoiling parts on the gun mount about the hinge A,

lr-x cos 0 1 - x cos 0
Qt(xt* + d tan 0)-Q2( +d tan 0-x )

oo: C°3 0 WxiEWUAiU^aaaal cos * ?bi?ib
+uQ (d+y )+uQ (d-y )+Bdh= 2M_Q (4)

1 " 1 2 2 D ~ a

but uQiyi-uQzya=B(d-db)+Pe-Qixt-a8x2
and uQ <-uQ =R

1 2

Substituting these values in (4), we have,

lr-x cos 0

(Qt-Q2)( + d tan 0)Bd+Rd+Ae=SMra

cos 0

that is, -Wr(lr-x cos j0)-Wr sin 0 d+Bd+Fe+Rd=»2Mr
or simplifying and combining, we have,

(B+R-«lr sin 0)d+Fe-Wr(lr-x cos 0)=2Mr (5)
at maximum powder pressure, x is usually negligible
and the equation reduces to:

(B+R-Wr sin 0)d+F9-Wrlr= SMra (51)

From this we observe that the reaction between the
recoiling parts and the mount is equivalent in effect
to a force (B+R-Wrsin 0), the line of action of which
is parallel to the axis of the bore or guides and

134

passes through the center of gravity of the recoiling

parts, and a couple of magnitude Fe, due to the powder
pressure, together with a component equal to the weight
of the recoiling parts and in its line of action assumed
concentrated .

Thus the reaction on the gun mount of the recoil-
ing parts, therefore, is equivalent to a single con-
centrated force, the resultant of (3+R-Wr sin 0), equal
to the total resistance to recoil and a force equal
to the weight of the system together with a couple Fe.
Since a couple and a single force in the same plane
are equivalent in effect to a single force, parallel
to the former, and displaced from it equal to the
couple divided by the force, the resultant reaction
on the mount of the recoiling parts reduces to a single
force; the resultant of B+R-W sin 0 and Wr, which be-
comes, since B+E-Wr sin 0 = K, equal to

J= /K2+yf* -2KWr sin 0

and the line of action of J makes an angle

-l(Wrcos 0) -KW-cos 0)
& =Tan *- =Tan *

(K+Wrsin 0) (B+R)

with the axis of the bore and is displaced a dis-
tance Fe. frora the center of gravity of the recoiling
J

parts. It is however, more convenient in computation

to resolve this resultant into its components, K and
Wr together with Fe.

If now we consider the equilibrium of the gun
carriage mount, we have for moments about the hinge
point, 2Mra- WaLa +Vb 1=0 that is,

(B+R-Vfrsin 0)d+Fe-Wrlr -WaLa+Vb 1 = C (6)
and since Wala + Wrir = WS18

135

136

Equation (6) reduces to

(8+R-Wrsin 0)d+Fe-WsLs+Vb 1 =0 (6')

which of course is the same as the equation obtained

in Chapter III.

It is, however, very often more convenient to
regard the mutual reaction between the recoiling
parts and carriage as divided into component .re-
actions along and normal to the axis of the bore to-
gether with a couple. See fig. (3).

Let x and y be the coordinate axes along and
normal to the axis of the bore or guides.

Xr= the sum of the component reactions along the

x axis.
Yr= the sum of the component reactions along the

y axis.

Now by introducing a couple Mr between the re-
coiling parts and carriage, it is entirely immaterial
where we assume the line of action of Xr and Yr.

Let r = the perpendicular distance from Xr to the

center of gravity of the recoiling mass.
z = the perpendicular distance from Yr to

the center of gravity of the recoiling
mass.

Considering the equations of motion of the re-
coiling mass, we have,
,2
P*wrsin /HCr=Mr~j-

Yr»Wrcos 0 ) (7)

» Pe +Xrr -Yr z

-

But Xr-Wrsin 0 = P-Mr 777 =Ka is the resistance

to recoil during
the accelerating period, and 2

Xr-Wr sin 0 = - Mr -jpr = Kr ia the resistance

__ to recoil during

the retardation period. In general Ka and Kr are

137

Fig. 3

138

different in value. Hence, let K = Xr~Wrsin 0 for

any given displacement x of the recoil.

If we now consider the reaction of the recoiling

parts on the carriage proper, the moments about the
binge A of this reaction, becomes,

lp- x cos 0

Xr(d-r)-Yr( + d tan 0-z)+Mr=ZMra

cos 0

Inserting values for Yr and Mr from the equations of
notion of the recoiling parts, we have:

Xrd-Wrlr+Wrx cos 0-Wrsin 0d+Pe = 2Mra

hence,

(XrWrsin 0)d-Wr(lr-x cos 0)+Pe= 2Mra )

or (

Kd-Wr(lr-X cos 0)+Pe= SMra )

which is the same expression as obtained before.

We may regard the line of action of Xr and Yr
to pass through the center of gravity of the re-
coiling mass, together with a couple M=Pe, the
powder pressure couple about the center of gravity
of the recoiling parts. See fig. (4)

Taking moments about A we have directly

1 - x cos 0

Xrd-Yr( ) + d tan 0)+Pe = 2Ura (8)

cos 0

and since Yr= Nr cos 0. we have as before

(Xr-Wrsin0)d+Pe-Wr(lr-x cos#)= 2Mra (9)

where JCr=B+R

The object of this analysis has been to show that
so far as the external effect of the reaction of the

recoiling parts on the carriage mount is concerned
the exact location of rod pulls or the lins of action
of the guide frictions, is entirely immaterial, though
as we shall see immediately, the value of R, the sum
of the guide frictions, does depend upon the line of
actions of these pulls together with the friction line

139

140

of action of the guide friction itself and thus in-
directly the external effect on the carriage mount is
affected slightly.

Further the location of the center of gravity of
the recoiling parts may considerably change the guide
frictions during the accelerating or powder pressure
period.

BRAKING PULLS The total resistance to recoil
if assumed constant throughout the
recoil is readily evaluated from the
following relations:
If K = total resistance to recoil

(assumed constant) (Ibs)

b = length of recoil (ft)

Vf= velocity of free recoil at end of powder period

(ft/sec)
B = displacement of free recoil at end of powder period

(ft)
T= time of powder period (sec)

Then from the energy equation for the movement of
the recoiling parts after the powder period, we have,

(ft.lbs)

Simplifying,

K = « mr f (Ibs)

b-E+VfT

With a variable recoil consistent with a stability
slope "m", and assuming a constant resistance during
the powder period, we have,

If K= the resistance to recoil in battery

k= the resistance to recoil out of battery

instability slope = °"r

h (h=height of axis of

bore above
ground)

141

2

then :^l{b-(E- —-)]= £ mr(Vf-~)* (ft. Ibs)

m
and k =K-m[b-(E- - — )] (Ibs)

Combining and simplifying, we have,

K = ~L^~ ~* (lbs)

2[b-E+VfT- - — (b-E)] in battery

£ fflp

at ettikjj si :£!!•.•<> r»i V"^^*^09

KT2
k =K-m(b-(E- - — )] out of battery

II ) , r •

B-the total braking pull (Ibs)

R=the total guide friction (Ibs)

Pk=tne total oil pressure on the hydraulic pis-
ton (Ibs)

p'

rj)=the hydraulic reaction plus the joint frictions

(stuffing box + piston) (Ibs)

Pa=the total elastic reaction(due to compressed

air or springs) (Ibs)

p'

ra=the total elastic reaction plus the joint

frictions (stuffing box + piston) (Ibs)
fti=the normal front guide reaction (Ibs)

^»=tbe normal rear guide reaction (Ibs)
u=coef f icisnt of guide friction (0.15 to 0.25)

Then K=B+R-Wr sin 0 (Ibs) Total resistance to recoil
where

B=P' + P' (Ibs) Total braking
h a

R= u(Qt+Q8) (Ibs) Total guide friction

The stuffing box friction is usually assumed at
from 100 to 150 Ibs. per inch of diameter of rod, and
if du and da are the stuffing box diameters of the

hydraulic and air cylinders respectively, we have

142

(Ibs)

Pa=Pa+100da (Ibs)

GUIDE OR CLIP REACTIONS The recoiling mass is

constrained to translation

V <• C 1 1

parallel to the axis of the
jc. 3 (i r

"bore by the recoiling masses

engaging in suitable guides in

the cradle of the top carriage. In general the re-
coiling mass may recoil in a sleeve, a part of the
cradle, or along guides considerable below the axis
of the bore and the center of gravity of the recoiling
parts .

For the former case, considering the external re-
actions on the recoiling mass, fig. (5)

aa-at=Wr cos 0

Q xl + &8x2+u (Qtyt-Q-2y2)-Fe-B b=0 (moments about

the center of
gravity of re-
coiling parts)
where eb= d-db , then &txt+(at+llr cos 0)+u (Q^-Q^y.,-

cos 0)-"Fe-Bb = 0
cos J0(x2~uy2)- Fe-Bb=0

Hence Fe+Beh-W_ cos 0(x -uy )

Qt= — * (10)

Further

Fe+Beh-W_ cos 0 x +W_cos 0 uy +Vlrcos 0 x +W_cos 0 x +

U* 2* 2' 1* 2

Wr cos 0 uy -Vf.cos 0 uy

i 'II "2

143

0(x +uv )
Hence Q2= - - - — (11)

When the guide reactions are below the axis of the bore
as in Fig. (4) y2 remains the same in the above formulae
whereas yt reverses in sign. Hence for case (2)

Fe+Beb-W_cos 0(x -uy )
Q= - — - - - - 2_ (12)

and Fe+Beh+HLcos 0(x -ay )

ft= - loti - * - L_ (13)

The total guide friction becomes, R=«(Q1+Q8)
hence

2(Fe+Beb )-Wrcos 0 x2+Wrcos 0 ny2+ ¥rcos
xt + x2 + u(yt - ya)

2(Fe+Beh)+Wrcos 0f (x -x J + ii(y +y )]

(14)

x +x +.u(y -y )

1 2 w 1 ' S

Now if M=x1-t-x,+ u(y1-y,!) for case I
or M=x1+xs-u(yi+ya) for case II

and if N=(xt~X2)+ u(yt+ye) for case I
N=(xi-x2)+u(y2~y1) for case II

we have therefore, in general that

2(Te+Beb)+Wrcos 0 N

R= - - - - - u (15)

M

which gives the total guide friction. The value of
the coefficient of friction u ranges from 0.15 to
0.20.

The total braking evidently becomes,

2(Pe+Beb)+Wrcos 0 N
B+ - u-K+Wr sin 0

M

or B(M+2uh)=(K+Wr sin 0)M-(2Pe+Wrcos 0 N)u
hance (K+Wrsin 0)M-(2Pe+Wr cos 0N)u

BS .

M+2Uh

which gives the total recuperator reaction in terms
of the total resistance to recoil.

Denoting as before by

Pjj= the hydraulic reaction plus the joint frictions

(stuffing box and piston)
P0 ~ the total elastic reaction plus the joint

3

frictions.
6^= distance from center of gravity of recoil-

ing parts to line of action of hydraulic

brake pull Pn
ea= distance from center of gravity of recoil-

ing parts to line of action of the re-

cuperator reaction Pa
The front and back clip reactions become,

, a a r ,g (

and

cos

(17)

y^ reversing in sign when the guide reactions are
entirely below the axis of the bores. Combining
as before and noting that F=u(Q +Q ) we have

t 2

2Fe+22p' ea+2ZPu eh+Wr cos 0 N

R- — h h r (18)

M
where M and K are the constants referred before.

Now K= £Pa+2Ph+R-Wr sin 0 (19)

145
and combining (18) and (19) we get

KM=(2Pa+2Ph -Wrsin 0)M+ n(2Fe+22Paea+22Pheh+Wrco5 0 N)

Simplifying,

KM=((2Pa-Wrsin0)M+ZP,!| x M+ u(2Fe+22Paea+Wrcos 0

or further simplifying,

a+Wrcos 0 N)
Hence, M(K_sp'+Win 0)-u(2Fe+2XP'ea+N Wnos 0)

I at 0 a

M+2U6J,

which gives the gross hydraulic pull in terms of the
total resistance to recoil, the gross air or spring
reaction and the maximum powder force.

APPROXIMATE FORMULAE Assuming the reaction be-
GUIDE FRICTION tween the recoiling parts to

be equivalent to a normal
force passing through the
center of gravity N, a couple

M, and the braking and guide friction forces B and R
having moment arms about the center of gravity of the
recoiling mass equal to dv and r respectively where
r is the mean distance to the guids frictions, we
have, for moments about the center of gravity of the
recoiling parts,
Be^+Rr=M neglecting the powder effect which is

usually very small and N=Wr cos 0 for the
total reaction.

Obviously the actual normal guide reactions,
becomes,

0 x

and . W_ cos 0 x.

fl s -M+ i.

* 1 1

where 1 = x +x also R=u(N +N )

12 12

146

2M+lfrcos 0(x -x )
hence R=u ( • — )

Substituting the value of M, we obtain,
u Wrcos 0(xt-x2)

(21)

l-2ur

Very often xl~xi is small and in a preliminary design
xt nay be assumed equal to xe
Hence 2ufieb

: l-2Ur (22)

which gives an approximate value of the guide
friction, useful in a preliminary design - u may be
assumed from 0.13 to 0.25.

Very often as in symmetrical barbette mounts,
the value of Beb may be small due to a small value
of eb and a certain limitation arises as to the
use of the friction formula previously derived.

When lf cos 0 x *

1 1

that is, Wrcos 0 xf = Be0+Rr = 8eb approx. we have

continuous contact along the guides, the distributed
guide reaction oalancing the weight comoonent normal
bo the guides.

For such a condition the guide friction,
becomes,

R=0.2 to 0.3Wrcos 0 (23)

INCREASE OF GUIDE FRICTION If we assume the total
DURING POWDER PRESSURE oraking B to be constant
PERIOD. during the powder pres-

sure period, the guide frict-
ion R is augmented by the

powder pressure couple together with the increased
friction couple.

Let B = the constant braking force
fa the varying powder force

147

Nt= the normal reaction of the stationary part

on the recoiling mass.
R = the guide friction during the powder pressure

period.
M = the reacting couple of the stationary part

upon the recoiling mass.
RS and M2 are the corresponding values during the

retardation period.
Then, during the powder pressure period, we have

N-Wr cos 0=0 (24)

and during the retardation period, we have,
-Wrsin 0=Mr

d'x

dt«
N-Wr cos 0=0 (25)

Further, let AM=M1-M3 and AR=Rt~R2 , then subtracting
(25) from (24) we have AM=Fe+AR r (26)

Now during the accelerating period the normal
guide reactions, become,

, Mt Wrcos 0 xf

(27)

U ' - *«- KrCOS 0 Xi

and during the subsequent retardation

142 Wrcos 0 x?

W*= T 1 (28

M2 Wr cos 0 x i

Na= T~ "T~

148

Adding the two equations in (27) and (28) respect-
ively and subtracting (28) from (2?) and multiplying
by the coefficient of friction n, we obtain obvious
expression:

2AM

*"• T U (28)

Substituting (29) in (26), vie have

AM=Fe+ ur (29)

Fe Fel
and AJ4- - = -

1_ iHE 1 -2ur (30)

1

and substituting in (29), ire have for the change of
friction during the powder period,

(31)

Thus the guide friction is continuously augmented
always proportional to the total powder pressure,
providing the braking is assumed constant. We also
note an additional cause of first class importance
for the reduction of "e", that is, the importance of
locating the center of gravity of the recoiling mass
along the axis of the bore.

Another cause for a change in guide friction during
the powder period is due to the torque reaction of
the rifling, Tr though the total guide friction remains
the same.

The normal reaction on the left guide, becomes,

m M_ wrcoa 0*a Tr
Ntl3 2, " 2dg

(32)

149

and the same for the right guide, becomes,

U W-COS 0X., Tr

N s * E 1 + -£_ (33)

*r 2X 2l 2dg

Wrcos 0xt Tr
2l ' 2dg"

where dg is the distance between guides.

In the gun recoiling in a sleeve this torque
must be balanced by the reaction of the key way.

Noting that, R-w(Ntl*Ntl + N +Htr)»u(Nt + Nt)

the total friction remains the same.

It is important to note that the friction on the
left guide over that on the right due to this rifling
introduces a couple in the plane of t"he guides which
tends to cause rotation about an axis normal to the
plane of the guides. Therefore, it is always essential
that small side grooves or flanges on the clips be
introduced. The additional friction on the flanges
is entirely naglsgible, but the normal reaction to
the flange in extreme cases may be considerable.

During the recoil the guide friction is seldom
constant since the distance between clip reactions pro-
gressively decreases in the recoil, that is the front
clip approaches the rear part of the guide in recoil.
When the recoil is long it is desirable on field carriages
to have an additional clip near the muzzle which engages
in the guide sometime later in the recoil. Due to this
cause the guide friction continually increases until
the engagement of the outer clip and then we have a
sudden drop in the magnitude of the friction.

When the braking pull remains constant and the
powder pressure coupls is small and n o outer clips are
introduced during the recoil, the clip reactions should
always be designed for the condition of out of battery.

To recapitulate in the limitations in design so far
as guide is concerned, we note,

150

(1) The bearing pressures and consequent
friction of the guide are reduced by increas-
ing the distance between the clip reactions
nearly directly, consequently for a given
guide reaction and friction, we have a
minimum distance between clip contacts

on the guides.

(2) The guide reactions are reduced by

bringing the resultant of the rod pulls
through the center of gravity of the
recoiling parts.

(3) The moment effect and consequent
guide reactions are further reduced by
bringing the resultant guide friction
line through the center of gravity of
the recoiling mass.

(4) It is highly desirable to center the
center of gravity of the recoiling mass
midway between the guide reactions.
This condition is usually impossible to
attain especially out of battery, but
may be compensated by increasing "1"
the distance between the clips, by an
additional front clip near the muzzle.

(5) Proper functioning of the recoil
may be srrtirely destroyed by having the
center of gravity of the recoiling mass
too far below the axis of the bore, thus
introducing a powder pressure couple
with excessive guide friction during the
powder pressure period. This powder
pressure couple may cause a "springing"
of the guides and considerable heating
as well. The center of gravity of the
recoiling mass should never exceed 1.5"
from the axis of the bore unless a friction
disk for rotation during recoil about

the trunnions is introduced.

151

COMPUTATION OP BRAKING We have seen from the
POLLS previous discussions that

the guide friction is not
independent of the braking
pulls due to the hydraulic

and recuperator reactions. These pulls tend to cause
rotation and thus augment the guide friction over
that due to the weight component.

The total resistance to recoil is given by:-
(1) when constant during recoil,

(lbs)

b-E+VfT
where Vf= maximum free recoil velocity (ft/sec)

T = total powder period (sec)

E=free recoil displacement during powder

period (ft)

b= length of recoil (ft)
•r= recoiling mass

(2) when variable consistent with the
stability slope "m",

mrVc+m(b-E)2
K= - ; - (Ibs)

2[b-E+VfT- ^— (b-E)]
2 mr

where

K=B + Rg-Wrsin0=Pn+Pa+u(Q1-»-Q2)-Hfrsin0 (Ibs)

and 8=Pn+Pa= total braking

Rg=u(Ql+Q9)= mean guide friction assumed constant

2uBeh
We have seen -— Ibs.approx.

where nf=0.15 to 0.2

e)j= distance from center of gravity of recoiling
parts to line of action of B. (in)

162

B3 total hydraulic and recuperator pull (Ibs)

I3 total distance between clip reactions (in)
r= distance from center of gravity of recoiling

parts to mean friction line (in)
then,

2uBeb

' - " (Ibs)

hence,

(K+«_sin 0)(l-2ur)
B« T - -
l+2u(eb-r)

Further since,

(Ibs)

8in

we have on simplifying
~~J"" (K+Wrsin j0)(

(Ibs)

l-2u(eb-r)

Very approximately,

Rg=0.3 Wr cos 0
and

B=K+Wr sin 0-0.3 Wr cos 0 (Ibs)

Pn»K+Wrsin 0-Fa-0.3Wr cos 0 (Ibs)

INCREASE OF RESISTANCE TO During the powder
RECOIL DURING POWDER PERIOD period, the powder pres-
sure couple may be suf-
ficient to cause a large
increase in the guide

friction, whereas the braking pulls due to the hy-
draulic resistance and recuperator reaction are not

153

affected. Prom the previous discussion on guide
friction, the increment in guide friction equals,

2Feu

(lba)

or more exactly

2Feu
2 M
where

M=xl+x2+u(yt-ys) Guides above and below

axis of bore

=xt+x2-u(yt+y2) Guides entirely below

axis of bore.

Hence the total resistance to recoil becomes during
the powder period,

K*=K + ARg (Ibs)

and this value should be used in the computation of
the trunnion and elevating gsar reactions.

Strictly speaking, the value of K is slightly
high, since the augmented friction due to a large
powder pressure couple, will diminish the maximum
velocity of restrained recoil and thus the resistance
to recoil for a given displacement b.

A more exact value of the resistance to recoil
can be estimated as follows:
Thus,

(K+ARtf)T* i (K+ARff)T

K[b-(E — )]= - mr[Vf * — ]

2mr 2 mr

Simplifying, we have,

* r f *' J 2m_

R. ?— (Ibs)

154

•bore 2uPae 2nFme

£Rtf» ————— - - (approx)(lbs)
"»+x+u(-y) l-2ur

(lbs)

Fro« interior ballistics, we have,

' w aVu

P«*« 1.12 -

« (b'+u0)8

w = weight of projectile
(Ibs)
uo- total travel up bore

(ft)
v = muzzle velocity (ft/sec)

b'-u0[(|l-*-l)±/(l-ll-)a-l] <">

~ (ft/sec)

u

where P,= total maximum powder force (Ibs)

Unless the powder pressure couple is excessive,
that is the center of gravity of the recoiling parts
is considerably below the axis of the bore the above
refinement in calculation is unnecessary. When e
exceeds 1.5 to 2 inches the above effect becomes of
consideration.

INTERNAL STRSSS IN THE It is very important to
RECOILING PARTS observe that the braking

force 8 when treating of the
external forces on the re-
coiling masses as in the

previous discussions refers always to the reaction of
the oil in the hydraulic brake and the spring or com-

155

pressed air reaction of the recuperator. During
the accelerating period the reaction on the gun lug

nay differ considerably from the braking force B due
to the acceleration of the piston and rods where these
recoil with the gun or to the acceleration of the re-
cuperator sleigh or slide when the sleigh recoils and
the rods are fixed to the carriage.

If now we consider a recoiling mass consisting
of a gun together with a single cylinder recoiling
with the gun, figure (4) and if we let,

B3 the total braking force along the axis of the

cylinder.
B =the normal reaction of cylinder on the gun lug.

j =the tangential or shear reaction on the gun lug
M -the bending moment reaction on the gun lug.

Neglecting the guide friction, let,

QI and QS be the normal guide reactions

xt and x? the coordinates along the axis of the
bore of the clip reactions with origin
at the center of gravity of the gun.

x and x the coordinates parallel to the axis

of the bore with origin at the center
of gravity of the recoiling parts.
xc and yc=the coordinates of the center of

gravity of the recoiling cylind«r with
respect to the center of gravity of the
gun as origin.

e^- distance from the center line of the recoil
cylinder to a line through the center of
gravity of recoiling parts parallel to the
axis of the bore.
.Mr and Wr= mass and weight of the recoiling parts.

Mc and tfc* mass and weight of the recoiling cy-
linder.

j = the distance from the shear reaction on the
gun lug to the center of gravity of the gun

156

itself.

F» the maximum powder force along the axis of the
bore.

If the mass of the lug is negligible as compared
with the mass of the gun, the coordinates of the center
of gravity of the recoiling mass with respect to that
of the gun becomes

Vc Vc
xr» -r — and yr=» e= — —

"r "r

Further

*t - - - and y y - -^j—
" "

«.- s * — and *. 3 ^

Now considering the gun with its lug alone, the
reaction of the recoil cylinder on the lug, consists
of the pull B a bending moment Wc cos 0 (xc + j ) and
a shear reaction Wr cos 0.

Taking moments about the center of gravity of the
gun, we have fi'(b+e) * \»c cos A j-Wc cos 0(xc+j )aQ8x^*

B'eb*B'e-Wcco. 0 xc=Qt(xt+ -~^)*(ar

wr wr

Simplifying, we have,

B'eb+B'e = Ot(xt+xt) + Wr cos 0 xa

Considering the recoil cylinder alone we note
that during the accelerating period,

d'x

B -B+Wcsin0 »MC -
° df

157

but from the recoiling mass, we have,
d*x F-B+Wr sin 0

dt« Wr

hence B' = B-Vfc sin 0 + r-(F-B+Wr sin 0) and substituting

in the previous equation, we have,

V
(B-Wcsin 0 + --(F-B+Wrsin 0)(eb+e)=Qt

"r

"r5
but yc= 8t>+« and yr= e hence (ejj+e)= - —

'r

Wre

"c

Therefore, substituting in the above equation, we have,
8(eb+e)-Wre sin 0+Fe-Be+Wrc sin 0 = Q1(xt+xt)+Wr cos £) ;
hence Beh+Pe =Q (x +x )+W_ cos 0 x.

Obviously if we consider the recoiling mass, fig. (4)
we have, taking moments about the center of gravity
X Qx tut Q-Qs W cos A

Hence Be^ + Fe - Qt(xt+ xf) + Wr cos 0 xa the same

equation as obtained above as of course we should ex-
pect. The above discussion shows the importance of
considering either the mass of the gun with its proper
external reactions or the mass of the recoiling parts
with its proper external reactions and not confusing
the mass, of the gun and recoiling parts, and the co-
ordinates of their center of gravities.

The maximum stress in a section m - m, see fig. (4)
of the gun lug obviously occurs when the bending moment
due to the weight of the recoil cylinder is a minimum and
the braking force B a maximum that is at maximum elevation,
In the above discussion the normal reaction between the

158

piston surface and cylinder was assumed zero. This
reaction obviously depends upon the weight and
relative deflections of the rods and cylinders. If
these weights were equal and at the same distance from
the point of support, and with equal elasticity,
this reaction becomes zero and we have the bending
moment assumed; but since the rods are relatively very
elastic as compared with the cylinder in general the
moment Wc cos 0 (xc + j) if anything is augmented.

If Imn is the moment of inertia of the section
"« - m", AJJ-U its area of cross section, and y is

the distance to the edge from the neutral axis of
the section and "g" the distance from 6 to the
neutral axis, we have for the maximum fibre stress

cos 0(xc tJ)]y fccos 0

+ - (34)

c

where B =B-Wcsin 0 + — (F-B+Wr sin 0)

wr

Since the weight components are small as com-
pared with the powder pressure force and braking for
a first approximation, we have,

wc

[B+ —-(P-B)lg
Wr

_

a-n>= (35)

lm-m

which is a useful formulae for practical design.

TIPPING PARTS The tipping parts consist of

all the parts that move in elevation
with the gun. The two principje
parts of the tipping parts are the
recoiling parts and cradle, the
one moving in recoil and the other remaining
stationary. The cradle supports by its guides the

159

recoiling parts on recoil, it takes the reaction of
the braking exerted on the recoiling mass and is sup-
ported by trunnions resting in bearings in the top
carriage and is further prevented from rotating about
these trunnions during the recoil by the reaction
between the elevating pinions of the top carriage and
the elevating arc of the cradle. When a rocker is in-
troduced between the elevating pinion and cradle for
an independent line of sight it should, properly speak-
ing, be included in the tipping parts.

It is of fundamental importance to always balance
the center of gravity of the tipping parts about the
trunnion axis since with massive parts the elevating
process must be done quickly and with the minimum re-
action on the elevating pinion of the top carriage.
Let x and y = the coordinates parallel and nor-
mal to the axis of the bore.

X and Y = the x and y components of the trun-
nion reactions.

F = the total powder pressure force.
E = the reaction between the pinion and

elevating arc.
j 3 the radius of the elevating arc.

6ea the angle between the "y" axis and
the radius to the elevating pinion

contact with the elevating arc.

The mutual reaction between the tipping parts and
'top carriage may be divided into the component reactions
X and Y of the trunnions and the elevating arc reaction
E.

By D'Alerabert's principle, considering the inertia
of the recoiling mass as an equilibriating force, we
have during the powder pressure period assuming the
gun practically in battery, for equilibrium of the
tipping parts, that, fig. (5)

,t

F-Mr — ~ -2X +Wt sin 0 +E cos Ge * 0 (1)

d u

for motion along the "x" axis,

160

9 ~.s?.} no tsdiexe fco

-;*»oq
for motion along the "y" axis, and ^ »Mt,w

d*x
bnrp(® + s)-Mr J^T 8-E j = 0

For moments about the trunnions, the weight of
the tipping parts having no moment since the

center of gravity is at the trunnions in battery.

1 i • • •

d*x

But F-Mr T~T =Ka fcns Votal resistance to recoi|.,T

*pj ?. 1 -,..-, .dur ing th8 accelerating period.
Hence equation (3) reduces to: Pa + K.s-E j = 0 and DO^
the reaction on the elevating arc in battery becomes,

a ( 4 V '

J »fU lc eixs 5dJ p4
and the trunnion reactions in battery, becomes,

(Fe+Kas)
2X=Ka*Wtsin 0 * cosee ) •

(
fcn* nointq edJ ne>ev?ift<j isoiJoaci edi « 3 v

(Fe+Kas) ty ^ (5)

2Y*Wt C03.0- "~~~ ' sin 9 (

tb« resultant trunnion reaction

•y

"

S* / X*+Y* making ^tn angle tan"1 — with the
"x" attWfc now if, fig. (5)

"rf^ =-'!l-0tal weight of the tipping parts
Wc = weight of the cradle :6f the tipping parts

Wr = weight of the recoiling parts
3.T* 3d ;
. It * fche horizontal distance to the center of

•

gravity of the tipping parts (in battsry)

always assumed at the trunnions from the

' • i. f~ ! u r A n y §

hinge point of the top carriage.

• To account for contact of tooth rack with the pinion
or worm of the elevating gear, the reaction B makes an

gle approx. 20° with the tangent to pitch lines,
erefore in above equations (eg. 5) *J • beoome«,"j o
• and "oos Ce" beooaes » oos 1C +20;" and "«in Q "

angle

Therefore

20

beooaea •«in(«t+20)

161

fcoJrqqii

: Js si

L-___i_i!L:_iW

*+.

iio^i <?rfi lo v

*— — "}' - V Oi ^

s s|*'i v^r

. 5

162

1^ * the horizontal distance from hinge point

of the tipping parts when the recoiling mass

is at a distance "XH out of battery.
lc = the constant horizontal distance from hinge

point of the center of gravity of the cradle.
Then for moments about the hinge point in battery,

"t^t* "r^r* "c^c and for a displacement "x" of
the recoiling mass from the initial position, we have,
WtlJ = Wr(lr - x cos 0)+«clc Therefore, the moment

of the tipping parts for a displacement "x" of the re-
coil, becomes,

Wt(lt - lj) »Wr x cos 0) (6)

Hence, for any position out of battery of the recoiling
•ass, i

2X-MP 2-7 -Wt sin 0 -E cos 9e» 0 )

( (7)
2Y-Wtcos 0+E sin 6e = 0 )

for motion along the x and y axis, respectively, and

t(r — S+»r x cos 0 -E j = 0 (8)

dt»

for moments about the trunnion axis. But the total re-
sistance during the retardation becomes,

.»
Kr-Mr f -B+R-Hr sin 0 (9)

Combining and reducing, we get for the reaction on the
elevating arc, for a recoil "x"

Kr s+Wrx cos

and the trunnion reactions for a recoil "x"

(10)

163

(Rrs+Wrx cos 0)

2X»K +Wt sin J6 + - : - cos ee ) -**'

J (

(Krs+Wr x cos 0) ) (11)

2Y=Kt cos 0 - - ; - sin ee (

J )

which shows the trunnion reaction depends only on the
total resistance to recoil and the moment effect of
the recoiling weight out of battery.

It is often more convenient to consider the reactions
of the elevating gear and trunnion reactions between
the tipping parts and top carriage as divided into
horizontal and vertical components rather than along
axis parallel and perpendicular to the guides.

The elevating gear reaction will be considered
positive when the line of action may be resolved into
components horizontally to the rear and vertically
upwards, that is when the radius joining the trunnion
to the elevating pinion contact with elevating rack
is measured from the vertical counter clockwise.
Galling this angle ne, we have,

9e= 0 + ne whereas before 0= angle of elevation.

We have then for the elevating components, measured
horizontally and not vertically.

He = E cos ne

Ve =E sin ne

and measured along the "x" and "y" axis, i.e. along
and normal to the axis of the bore,
X,» = E cos

YP=E sin(0+n_)

* More strictly to aooount for obliquity of tooth
contact of elevating me o h a n i 3 n , j beoor.es j oos 20,
003 0. becomes oos (W +20) and sin O becomes s i n ( 0 * 2O ) .

O o w

164

The horizontal and vortical components of the trunnion
reaction become, in battery,

(Fe+Ks)
2H=Ka cos 0+ — — - cos ne )

J

P +K
2V=Ka sin 0+Wr - — • - sin ne

J \

.v.ieJJed ito Ji ^ 9dj

and out of battery,

(Krs+Wrx cos 0)
2H»Kp cos 0+ - • - cos ne

fcnoi* narii ierij£-t feSnenoqaioo IfiaiJiev bnadfiJocsi-jorf
(Krs+Wrx cos 0) )

oi<" ; &•» erf ^s« noiJcs lo snil »riJ ae«

and the resultant trunnion reaction becomes,
no. "ot tuJ'fcsT sHJ aart* ai 3&di

S = j/p* + 7*

.*c

INTERNAL REACTIONS OF It is important to observe
TIPPING PARTS ~ that X, Y and E are the external
ROCKER INTRODUCED. reactions exerted by the top

carriage on the tipping parts
which include a rocker if used.

The total reaction on the trunnions include the reaction
of the top carriage X and Y and the reaction of the
rocker Xr and Yr . Hence the resultant reaction on th«
trunnions, become, algebraically,

l»oi

X =X+Xr )

and ((12) = the shear components of the

V =Y+Yr )

trunnion pins on the cradle.

The reactions on the rocker, alone, therefore, become,
the reaction of the top carriage pinion E, the reaction
of the trunnion Xr and Yr reversed and the reaction
of the cradle M reversed. In many cases an elevating
screw is used between the cradle and rocker and when
used, M reversed is the reaction of the elevating screw,

166

•~— — ^______^ -:i «•

10 fi>.7^nT~--

;-

fc{1 .* £fl * '5 « 5 j_/

,13JJuOf Sri-*

/ /

dd^ 1C 8S»4*M!ibT003 "^" bft* "x"

&sntiL'to , *•

. J.--

aQn««si& .! .-
*di to

' ^ »*<! *jV

t ** , -\ - -, - - - \ rl J

2 0R * 5l STdHw
•J «Ml* (.--

its line of action since the screw is hinge jointed
at its two ends necessarily lies along the axis of
the screw. The screw reacts on the cradle with a
force + M and is under a compression M during the
recoil.

Considering now the equilibrium of the rocker,

we have, if

•

Wr= the weight of the rocker

k= the perpendicular distance from the trunnions
to the line of action of M which is the line through

the axis of the elevating screw when used, or the
normal to the contact surface of rocker and cradle when
an elevating screw is not used.

B= the angle between the line of action of M and

the "y" axis.

xa and ym= the "x" and "y" coordinates of the
center of contact of rocker with
cradle, or hinge joint of elevating
screw of rocker on cradle.
xr and yr = the "x" and "y" coordinate of the

center of gravity of the rocker and
is measured from the trunnions towards
the breech and downward.

For equilibrium along the "x" axis (see fig. (6)

2Xr+E cos 6 + *r sin 0~M sin 8 = 0 (13)
for equilibrium along the Y axis

2Yr+M cos B-E sin 6C **r cos ^ = ° ^14^
and for moments about the trunnion,

Ej-Mk-W (xr cos 0-yr sin) = 0 (15)

where k = xra cos B+ym sin B. It is often convenient

to replace Xr cos 0-yr sin 0 =hr the horizontal distance

from the trunnions to the center of gravity of the
rocker, then (15) reduces to

167

Ej-Mk-Wr hr= 0 (15 ')

hence Ej_w' h

M= (16)

K

and

(Ej-w; hr)
2Xr= sin B-Wr sin 0-E cos ee )

k (

> (17)

(Ej-Wr hr) <
2Yr-E sin 9 -Wr cos 0 - cos B )

K

which shows the rocker reactions depend only on the
elevating arc pressure.

If now we consider the equilibrium of the tipping
parts not including the rocker, we have, fig. (7)

2J('=Kr+(Wt-Wr) sin 0+M sin B )

(
2Y'»(Wt-Wr) cos 0 -ti cos B ) (18)

and Krs+Wrx cos 0-V»' hr

M= — — — (19)

k

since the moment of (Wt~wr^ about the trunnions= -Wr hr
but now from equation (12), we have

2X=lfX'-2Xr )
2Y=2Y'-2Yr (

hence substituting the values obtained in (17) and (18),
we have,

* To account for tooth contact, more strictly, re-
olaoe j to j cos 0.; oos 0. to oos.(0 +20) and sin. 0 to

168

w-t

bnsqsb eueJ:49£Ai

won

. -. o ft. r i fit . "I

.« i * tn & ( Ot »

169

( 2X=Kr+(Wt-tfr) sin 0+14 sin: B-<H s&n B-W sin £J-E cos 6e)
( 2Y=(Wt,-wr) c^3 0"14 C50S ?~(? r*ae^-Wri cos 0^»CoQS B)
Simplifying these values reduce' tV the- fofrnrer Values,

2X=Kr+Wt sin 0+E co« ee

.(v.TSJtfid ni nu§) g^rseq.

2Y=Wt cos 0 -B sin'0e'
and further

sieriw n.'4o artj \o eujbfii J •. 1 * ,1

i i

Krs +Wr x cos 0-IL. hr . EjHlL hr

- — - - - - — • _

k k

.<asc "io asjibAf n*eM B H
and .therefore as. befor* ,0 floU36il9b e ^

.noi4*vele cBorclxfi* Je gaii ::Krs+Wr x cos 0

B= - . *>
-* - iiu^nof; gnt-rqe

thus checking the formulas derived for the rocker

reactions.

oa be;.. boil\£o s/iiiqe 8.Hj ni SfiiTqs tsriT
TRUNNIONS LOCATED AT In guns shooting at -hijjjh J
THE REAKr BALANCING^ r. ,elevAti<m s,uc.A ;^s; aqti-
GEAR OR EQUILIBRATOR.. .T.,7cair;cra=ft ^ttn^,: mortars, and

even howitzers, it is often
necessary to Ipcate the trun-

nions in the rear or near the breach of the gun in order
to prevent the breech of the gun from striking the
ground during the recoil when the gun is elevated. Ob-
viously it is impracticable to balance the tipping
parts about the trunnions without some sort of a
balancing gear commonly known however, as an equilibrator

An equilibrator should balance the tipping parts,
since the center of gravity of the tipping parts are now
displaced forward of the trunnions, at all angles pf
elevation when the gun is in battery. It consists
sometimes of a cam arc, a chain passing over the con-
tour of the cam arc and connected with a spring
cylinder oscillating about the trunnions fixed to the

top carriage to take care of the small deflection due

170

to the change of radius of the cam.

PROCEDURE TN DS3TGN OT B B U I L I B R AT OR :

Let Wt = weight of the tipping parts.

ht a horizontal distance from the trunnions
to the center of gravity of the tipping
parts (gun in "battery).
ro = equivalent radius of cam at horizontal

elevation.
rn = final equivalent radius of the cam where

the cam arc has turned through the maximum
angle of elevation = 0.
B = Mean radius of cam.

dn = deflection of spring at zero elevation.
d0 = deflection of spring at maximum elevation.
C = spring constant.

0 = the angle of elevation expresses in
rad ians .

The spring in the spring cylinder is arranged so
that it is in general under compression. As the gun
elevates, the compression of the spring is decreased
in virtue of the motion of the cam. The total motion
of the spring becomes,

R0 ' dh~do aPPro*' during the elevation 0 where

ro*rn

" for a first approximation and

m

c dhro= *tht for equilibrium about

the trunnions for
C dorn= *tht cos 0- all elevations.

If now we assume

dh=(- to -)d solid

11 3 4

and i t

do =(— to -g-)d solid

Then for a preliminary design, we have,

171

C = -i 5

(- to - )d Solid r0

Wtht cos 0
C =

* *

(- to -) d solid rn

r + r
and (- to -)d solid -(- to j)d solid = ( — - — )0

0

The unknowns in the above equatibns are C, d solid ro
and rn; hence if we assume any one of these values
the remaining values are determined.

The equivalent radius of the cam may be obtained
by a "point by point" method as follows'-
The initial radius, becomes, g h

"t L

r°*5*r

Now move the cam an increment angle A0 where nA0=0 the
total angle of elevation, and we have,

cos #

and Ad = rnA0

C(dh-r0A0)

then

W^ht cos 0

** C[d-(r0+rM07 >V<po+'

Htht cos 0
rn~ C[dh-(r0+rt rn_,)A0] dh~do= ro+rt~"rn-i)

A0

Strictly speaking the angle A0 in the above pro-
cedure should be augmented by the angle r"n~ r° — radians-

Where D = the perpendicular distance from the
trunnions to the extremity of the equivalent cam radius.

From the equivalent radius thus obtained the cam
contour may be drawn by drawing in a curve always
tangent to the

172

perpendiculars to these radius, drawn from their

f.
extremities.

With a balancing gear or equilibrator, the
trunnion reactions are modified and now become ___ - r
it

T = the tension in the chain.

a = the angle T makes with axis "X", (takea *) j
along the axis of the bore)

Ks*WrXcos 0+Fe

-a— iui)deos 0e*T cos • *>«3)»e-c
onj- \n& ;,,^ bns

>

)sin6e *T sin a iT(

J I v,d Jni:oq")B

and soihsT IsiJ/ni: ftri'JY

)

j
-f"' ei^ns-Jnsaaeioni ne

i!»8

Usually d = 0. and thus the Y component of the
trunnion reaction is unaffected.

DIRECT ACTING BALANCING Another form of

GEAR balancing gear, for nedj

balancing the tipping
'3+ 'O~ i,Parts at. all angles of <,

' elevation about the

trunnions which are located to the rear, consists
of a spring or pneumatic oscillating cylinder and
its rod directly connected between the tipping parts
and the top carriage.

In the position of the tipping parts at zero
elevation, gives maximum moment and therefore re-
quires the •axinua balancing reaction.

$rtj moil ocf.flJaib IB i • jdW

~ To .ccount for tooth contact of .levatl.g
••ohaniSB, replace j to j COS 20, oos S to oos
(Ot«20) »rd .In d, to .in(«e+20).

• n -

sri.t OJ

173

M ..-•? .noiievele ed.4 &i»jt«or> « ae eseeio

Wt = weiSnt °f th8 tipping parts

nt = horizontal distance from the trunnions tcbijr;nu}
the center of gravity of —the tipping parts -11^0
%nii el j(g un- i^ b *Vt &ry.<\» r, o x n n 01 4 eri J §fil,»f.!
.u^jfc *od yt - coordinates alon^ and normal to bore ;
-oneisd orjfiiiiuffroift- trunjiion to center of gravity

of tipping parts (gun in battery:)i.2.rq 5jni

0 - angle of elevation., Juov.*! VJ«nif«il9Tq s io1
0m = maximum elevation>.. erfj Ic sixs oornnot^ arii gni
r = radius from the trunnion to the crank pin
bft!j,llE:f*hich connects the tipping parts to the pi*-;

vfciJfcon rod of the oscillating cylinder, (in.) iBi^cni
R = reaction exerted by the balancing gear along
the piston rod of the oscillating cylinder;.

s) 3 iscfnos iiori TO Isi^iai siU :

dt = moment arm of R about trunnion. (in)

d = deflection of spring at horizontal elevation.

do = deflection of spring at maximum elevation.

(in) ,ooi**veJs uwsilxsa icl seasooed
C = spring constant.

RJ = initial, balancing reaction (0° elev.)
Rt = final balancing reaction (0* elev.)(lbs)
S = stroke of piston in oscillating cylinder. (in)
pt - final ,air piressure j.^ pneumatic balancing

cylinder (Ibs/sq.in.)
,|ȣ = initial air pressure in pneumatic balancing

cylinder (lbs/sq.in.1

^,f, effective area of balaacing pistoa ..(sjj.>cinj._) j
V0 = initial air volume (cu.in.)

*\L ,'J3bniI^o Jniiq* 6 ri; :?••
At any angle of elevation 0, we must have,
R dt » Et(xt cos J0-y-t sin 0) io?

In general the center of gravity of the tipping

parts lies approximately along the axis of the trunnions,
and therefore, R d = W x cos 0.

174

If dt remains constant, the reaction R should de-
crease as a cosinef unction in the elevation. Since it
is usually impossible to decrease R according to a cosine
function, we may so locate the trunnion of the oscillating
cylinder so that the product R dt = Wtxt cos j0. By
properly locating the trunnion axis of the oscillating
cylinder a very close balancing is possible throughout
the elevation either with a spring or pneumatic balanc-
ing piston.

For a preliminary layout, we may start by locat-
ing the trunnion axis of the oscillating cylinder
somewhere depending upon clearance considerations,
along a line parallel to the chord joining the assumed
initial and final positions of the crank and midway
betwean the chord and middle of arc. The crank turns

an angle equal to the total elevation &

m*

Then the initial or horizontal balancing reaction,
becomes,

r 0m

R^ - (1 + cos — r)=Wt xt and the final balancing
2 <c

reaction, becomes for maximum elevation,

r *m
Rf-(l+ coa T~-)3Wf (*t cos flfl+yt sin 0m)

3 Wt xt cos tm (approx.)

Rf
We have, therefore, cos 0m very roughly, and the

Ri *•

total stroke of the piston, becomes, S=2r sin -r— (in)

Now with a spring cylinder, if,

dh s 3 to '4 solid height of spring (in)

do = - to - solid height of spring (in)
then

dh-d0= S (in)

175

and at 0° elev. R^ = c dh at max. elev. Rf= c do.

Hence the required spring, may be approximated by
the solution of the following equations,

= = cos 0m )

dh-do='

21

Cdh=-

r(l+cos — ) (

•

With a pneumatic cylinder, we have, since the expansion,
may be assumed isothermal,

pf vo
—— = — — = cos J0jj, (aoorox.) from which we

i| vo+

may determine the initial volume Vo hence

pi cos 0^
V0 = AS = AS (approx.) (cu.in.)

*f 1- cos 0m

Now

pf

RiS TT

Ri = p^A and V0= — -— [ ] (cu.in)

Pi PP
1 --1

Pi

We see, therefore, to decrease the tulk of the
cylinder it is important to maintain as high an
initial air pressure as possible. It is reasonable
to assume the same initial pressure as used in the
recuperator brake.

Therefore, in the first approximation of the de-

176

sign layout of a pneumatic balancing gear, we may
start with,-

Pf

(cu.in)

V0 '

2WtxtS

'

PT

0 v

r(l+cos - )p;
2 ]

1 -

Pf

0 Pf

where S * 2 r sin - , — = cos 0m (approx.)
2 Pi

REACTIONS ON TIPPING PARTS With the trunnions to
WITH BALANCING GEAR. the rear of the center

of gravity of the tipping
parts and a balancing gear
introduced, we have a

cantilever form of top carriage, and the reactions on
the tipping parts are usually approxircataly in the
position shown in fig. (8).
Let
x and y - coordinates along and normal to axis of

bore.

R » reaction of balancing gear (Ibs)
9r= angle between R and y axis.
dts moment arm of R about the trunnions at any

elevation 0.

E = elevating arc reaction (Ibs)
9e= angle between the y axis and the radius to
the elevating pinion contact with the
elevating arc.

J= radius of elevating arc from trunnions (in)
X and Y = components along x and y of trunnion

reaction. (Ibs)
Wt » weight of tipping parts,
x^ and yt * coordinate of center of gravity of

tipping parts from trunnion (in)
Wr = weight of recoiling parts (Ibs)
He have, then, two positions to consider, the
in and out of battery positions respectively. In

177

JPEAC7/CW5 ON T/PP/MG PA/?TS W/77J
DIRECT ACr/N6 BALAMC/MG

TPU/W//OM AX/S

rig. a

178

the battery position we have the effect of the powder
pressure couple, while in the out of battery position,

we have the moment effect of the recoiling weight.
Considering the Recoiling parts in battery:

We have for the kinetic equilibrium of the tipping
parts, for motion along the x axis,

d%
Pb- mr — — -2X+Ht sin 0 +E cos ee + R sin 9r = 0

Q X

for motion along the y axis,

2Y-Wt cos 0 -E sin 6e + R cos 6r = 0

and for moments about the trunnion,

, 2
Pb(e+s)-mr— 7-S+R dt-Wt(xtcos 0 -yt sin 0)-Ej= 0

d t»

Since, however, 2

pb~rar = K and R d =

-y+ sin 0) requirecl condition of the balancing gear,
the above equations, reduce to,-

Ks+Pbe
E= - (Ibs) for the elevating arc re- )

action in battery (
2X=K+Wtsin0+E cos ee+R sin 9r (Ibs) )

2Y=lft cos 0+B sin 9e ~R cos er (Ibs) )

(

Wtxtcos 0 2Wtxtcos 0 )

where R = - - - = - - — approx.(lbs)

dt 0m (

r(l+ cos - )

It is to be noted that 2X, 2Y, E and R are to be
regarded as the reactions exerted by the top carriage
on the tipping carts, a rocker if used being included

* To account for tooth contact of elevating
••••••!••, substitute j cos 20 for j , B co»(C.-20)

for I cos O. and E sin(0 -20) for I iin 0..

09 o

179

as a part of the tipping parts. The resultant bearing
reaction between the top carriage bearing trunnions,
becomes,

S = / XZ + YZ (Ibs)
Considering the recoil parts out of battery

if

Wt= total weight of the tipping parts
Wc= weight of the cradle of the tipping parts.
Wr= weight of the recoiling parts.
xro and yro = battery coordination of the re-
coiling parts with respect to
the trunnions.
xc and yc = coordinates of the cradle with

respect to the trunnions.

Xfo and yfo = coordinates of the tipping parts
in battery with respect to the
trunnions .

For a displacement "x" of the recoiling parts
froB the initial battery position, we have, for moments
about the trunnion,

_To .SEC- I erii oJ v- rtiostu

d x

+WC sin 0 yc-E j = 0

U Utf v — W v +W v

POW n-fXj.1" ffT%A_»-.~n.>A-,

hence the moment equation about the trunnion reduces to
,»

mr - .t-txtcos +tytsn +rx cos 0-Ej= 0

QTf

but due to the balancing gear, we have,

"txt c°3 ~yi s^n

and further K=mr— - — — that is, the inertia resistance,
d t

equals the total resistance to

180

recoil.

Hence K s + Wr x cos 0

For motion along the x axis, we have, (
2X=K+R sin er+E cos ee+Wt sin 0 (

)

and for motion along the y axis, (

2Y=«ft cos 0+E sin e+g sin9s -R cos er )

I

where as before ;

2Wt xt cos 0
R= roughly. )

0o> (

r( 1+cos -— )

f )

From the above analysis, we see, therefore,
that the elevating arc reaction remains the same
with or without a balancing gear, while the
trunnion reactions may be increased or decreased
according to the location of the line of action of
the balancing gear.

INTERNAL REACTIONS OF The rocker reactions depend
TIPPING PARTS WITH solely on the elevating gear
BALANCING GEAR - reaction, and with a balanc-
ROCKER INTRODUCED. ing gear, the elevating gear

reaction is independent of the

eccentricity of the center of gravity of the tipping
parts from the trunnions. Therefore, the rocker re-
action on the trunnion is entirely independent of
the reaction exerted by the balancing gear or counter-
poise. In brief, the rocker reactions remain the

To account for tooth contact of elevating
•echanisB, substitute j cos 2O for ,i E oos(O.-20)
for B oos 68 and B Bin (O0 - 20) for B sin O°.

181

same with or without a counterpoise or balancing

gear. The reactions exerted by the top carriage
on the trunnion do however depend on the mag-
nitude and direction of the balancing gear.
Therefore, the shear and bending at the section
of the trunnion adjoining the cradle must also
depend on the balancing gear or counterpoise re-
action. .

An analytical proof of the reactions is given
as follows:-

Let

X and Y = trunnion components of the reaction
of top carriage.

Xr and Yr = trunnion components of the re-
action of the rocker.

X and Y = shear components of the trunnion
pins on the cradle.

Wr = weight of rocker.

E = elevating gear reaction on rooker

M = cradle reaction on rocker.

j = radius to elevating gear arc.

k = the perpendicular distance from the trunnions
to the line of action of M which is the line
through the axis of the elevating screw, when
used, or the normal to the contact surface of
rocker and cradle when an elevating screw is
not used.

B = the angle between the line of action of M
and the "y" axis.

xffl and ym = the "x" and "y" coordinates of the

cradle hinge joint of rocker elevat-
ing screw or the center of contact
of rocker on cradle.
xr and yr = the "x" and "y" coordinate of the

center of gravity of the rocker and
is measured from the trunnions towards
the breech and downward.

182

Evidently for the shear at the cradle section
of the trunnion,

x'= x + xr )

(
Y' = Y + Yr )

For the angular equilibrium of the rocker,
Ej-Hk-lfr(xr cos 0 - yr sin 0) = 0

if we let xr cos 0-yr sin 0 = hr, then

Bj-Wrhr
Ha 1 — , where k = xffl cos B + ym sin B, that is,

the cradle rocker reaction depends solely on the
elevating gear reaction.

For the translatory equilibrium of the rocker,
2Xr= M sin B-E cos ee -Wr sin 0

2Yr=E sin (Je ~Wr cos 0 -M. cos B

which shows the rocker reaction at the trunnion de-
pends only on the elevating arc pressure and there-
fore is independent of the counterpoise reaction.

Considering the equilibrium of the tipping parts
not including the rocker, we have,

2x' = Kr-i-(Wfc-1fr) sin 0 +M sin B*R sin 6r

2Y*=(Wt-Wr) cos 0-M cos B-R cos er
Further if measured from the trunnion axis,

lt=xtcos 0 -ytsin 0= the horizontal distance to center

of gravity of tipping parts(re-
coiling parts in battery)
lt= the horizontal distance to center of gravity of

tipping par.ts (recoiling parts out of battery)
lr= the horizontal distance to center of gravity of
recoiling Parts in battery.

183

lc= the horizontal distance to center of gravity of

cradle .
-hr= the horizontal distance to center of gravity of

rocker measured in a negative direction from the

i's.
d^ = moment arm of the counterpoise reaction R about

the trunnions.

Then since the moment of the tiooing parts minus
rocker about the trunnions is equal to the moment of
the weight of the tipping parts minus the moment of
the weight of the rocker, we have,
W~h=?1~ x cos

Now Ifrlr+Wclc= «Ttlt+Wrhr

hence Wtl^+Wrhr=Wtlt +Wrhr~Wr x cos 0

=Wt(xt cos 0-yt sin £J) + Wrhr~Wr x cos 0
therefore,
Krs+Rdt-Wt(xtcos 0-yt sin 0)-l?rhr+Wrx cos -Wk= 0

but for equilibrium of the tipping parts in battery
Rdt=Wt(xt cos 0-yt sin 0)

hence

Krs+Wrx cos 0-1
M = •

Since, however, 2X=2X'-2Xr )

2Y=2Y-2Yr

We have in substituting the previous values for
2X* and ZXp^j,

2X=Kr+Wtsin 0 +R sin 9r+E cos 6e
2Y=lft cos 0-E sin 9e~R cos 6r

184

Krs +Wrx cos 0
and E = —

J

In the preceeding analysis it is important to note,
that the center of gravity of the rocker is assumed
to the rear of the trunnions, and the elevating gear
reaction is considered positive when the radius to the
pinion contact of the elevating rack is measured
counter-clockwise with respect to the "y" axis
through the trunnions. Evidently when QQ is negative
(i.e. clockwise from "y" axis, E cos 6e remains the
same but E sin 6e becomes negative in the above
equations.

•ol'

EFFECT OF RIFLING Cue to the rifling, the
TORQUE ON TRUNNION torque exerted on the gun by
REACTION the shell aust be balanced in

considering the equilibrium
of the tipping parts by an

equal and opposite moment exerted by the top car-
riage on the trunnions(assuming due to the ouch
greater flexibility of the elevating arc and pinion
that the elevating arc reaction is entirely unaf-
fected).

If the rifling is right handed, then in the di-
rection of the muzzle, the Y component of the left
trunnion is increased and the Y comoonent of the
right trunnion is decreased by the amount equal to
the torque of rifling divided by the distance between
the trunnion bearings on the top carriage. Usually
this affect is quite negligible as compared with the
ther forces exerted.

STRENGTH OF THfe TRUNNIONS. The critical section

of the trunnions is usually
where the trunnion joins
the cradle.

Let "«n" represent this section on the trunnion,

see fig. (9).

185

xr.-s no

n

186

a = the distance from "mn" to the center of

the top carriage oearing.
b = the distance from "mn" to the center of the

rocker bearing .
MX = the bending moment at "urn" in the plane of

the X component reactions.
Hy = the bending moment at "mn" in the plane of

the Y component reactions.

M = the resultant B. M. on the section, "mn" .
f = maximum fibre stress.

D = diameter of the trunnions at section "mn"
I = the moment of inertia of the section about

diameter.

Then Mx= Xa+Xrb My= Ya+Yrb and M,/^J~"

further nD4 MD
I = and f =

64 21

32M 10.18M

hence f = — — = r—

nD D

Usually the fibre strass is limited from
— to — of the elastic limit of the material used,

2 3

and the minimum diarae ter of the trunnions becomes

/10.18M
D = 7

The shear stress is usually negligible as compared
with the bending stress.

LIMITATIONS ON THE EXTERNAL In considering the
REACTIONS OP THE TIPPING external reactions on
PARTS. the tipping parts, we

have to consider the
limitation imposed on

the elevating arc reaction and the reaction on the
trunnions by the top carriage.

187

ELEVATIHG ARC REACTIOH:

(1) The elevating arc reaction is reduced
by reducing the perpendicular distance
between the line of action of the re-
sistance to recoil, which passes through
the center of gravity of the recoiling
parts parallel to the axis of the bore,
and the trunnion axis. When the line
of action of the resistance to recoil
passes through the trunnion axis, the
reaction on the elevating arc in battery
is zero if we neglect the effect of the
powder couple and is equal to the moment
effect of the recoiling weight when the
gun is out of battery.

The elevating arc reaction is re-
duced proportionally to the increase of the
radius of the elevating arc.

The elevating arc reaction should al-
ways be considered in the limiting con-
ditions of in and out of battery, that
is, with the maximum powder pressure
couple acting and out of battery when
the maximum moment effect of the re-
coiling weight about the trunnions exists.

(4) When the resistance to recoil does
not pass through the trunnions the
elevating arc reaction due to the short-
ening of recoil is a maxim at max. ele-
vation.

TRUNNION REACTIONS:

(1) The "X" component of the trunnion
reaction (i.e. the component parallel
to the bore), depends upon the total
resistance to recoil and the component
of the elevating arc reaction parallel

188

to the bore as well as the weight
component of the tipping parts when the
gun elevates.

(2) The "Y" component depends upon the
weight of the tipping parts and the
component of the elevating arc pres-
sure parallel to the "y" axis.

(3) As the gun elevates the component

of the elevating arc pressure parallel
to the "x" axis decreases but the weight
component increases and due to the
shortening of recoil on elevating the
resistance to recoil increases.

(4) The component parallel to the "y"
axis of the elevating arc reaction
increases but in a negative direction,
thus tending to decrease the Y component
of the trunnion reaction. On high
elevation because of the large re-
sistance to recoil for a short recoil,
the elevating arc pressure parallel

to the "y" axis more than compensates
the decreased weight component of the
tiooin^ parts thus causing a reversal
of direction of the Y component re-
action of the trunnions.

(5) Thus in general the X component in-
creases while the Y component decreases,
verv often reversing on elevating the
gun and thus the trunnion bearing con-
tact may shift 90° or over.

STRESSES IN CRADLE OR The reactions on the
RECUPERATOR FORGING. cradle or recuperator are:
* (1) the trunnion reaction
of the top carriage on the
cradle: (2) the reaction

of the elevating arc which is equivalent to a single
force in the direction of the elevating pinion re-

189

action on the elevating arc together with an addition
al moment: (3) the reaction of the recoiling mass on
the guides: (4) a result and reaction parallel to the
longitudinal axis of the cradle or the guides due to
the various "pulls" exerted on the recoiling mass and
(5) a distributed load which is uniform if the cross
sections remain the same due to the weight of the
cradle.

In an accurate computation of the stresses in a
cradle it is necessary from a preliminary layout of
the cradle to locate roughly the neutral axis of each
section and connect these points for a longitudinal
neutral axis line. "We may then treat the cradle as a
simple beam, talcing into account the bending moments
caused by eccentric loads such as pull reactions off
the neutral axis, guide frictions, etc. The trunnions
usually are located considerably above the neutral axis
and the X component of the trunnion reaction causes a
large eccentric load with a consequent large abrupt
change in the bending moment diagram. This is usually
a characteristic in the bending moment diagram for all
cradles or recuperators using guides.

Let us now consider the various diagrams showing
the characteristics for bending moment, direct stress
and shear for the "Filloux" cradle as well as for the
"240 m/m Schneider Howitzer" cradle representing typical
cradles with guides (figures 10 and 11).

Neglecting the weight of cradle as relatively
small, and letting

Mt= max. bending moment at the trunnions.

Mc= max. bending moment at the point of contact
of the elevating arc with cradle.

Qt and Q? = the front and rear normal clip re-
actions.

x' and x' = the "x" coordinates of these re-

1 2

actions with respect to the trunnions,
d and d = the distance of the friction components

1 2

of Q. and Q from the neutral axis.

190

-»«-

SENDING

STRESS

155 Mr^n GUN CRRRIRGE:

MODEL OP 1916 FILLOUX
CRRDLE

SHE1RR

Fig. 10

191

B = the resultant of the "braking "pulls" reacting

on the cradle.

djj = the distance from the neutral axis to "B"
dx = the distance from the neutral axis to the

trunnions .

X and Y = the trunnion reactions on the cradle.
Xg and Yg = the elevating arc reaction on the

cradle .
Me = the moment exerted by the elevating arc on

the cradle.
= the distanc

, _^^

the distance from the neutral axis to Xe

xe = the "x" coordinate of the elevating arc

contact with respect to the trunnions.
The bending moment changes at the trunnions by
the amount 2X dx.

Now for guns with "braking pull" reactions on
the cradle in the rear as caused by the compression
of the oil and air in the recuperator as in the 155
m/m Filloux, we have for the bending moments at the
trunnions,

Mt = Qt(xt + udt) just to left of trunnion.
Mt = Qt(xt + udt) - 2X dx (just to right of trunnion)

and at the elevating arc
contact,

MC = V1 - xe * Ud8) -Bdb

As a check, we also have,

Mc = Qt(x^ + xe + udt) + 2Yxe - 2Xdx - Me

The bending moment MC is usually distributed caus-
ing a parabolic curve as shown in the B. M. diagram of
the 155 m/m Filloux..

For guns with "braking pull" reactions on the
cradle in the front, due to the tension in the
stationary hydraulic piston and recuperator rods,
as in the 240 m/m Schneider Howitzer, we have for
the bending moment at the trunnions

192

/"ft

"1 ' » P

J \

CRADLE

240 MM HOWITZER
SCHNEIDER
1916

BENDlNCa MOMENT

DIRECT STRESS

SHERR

rig. ii

193

1) -Bdj)( just to left of trunnions) and
Mt=Qt (x| + udt)-Bdjj-2X dx( just to right of trunnion)

Further Mc= Qt(x|*xc+udt )+2Yxe-2Xdx-Bdb or as a check
Mc=Gz(x^-xc+ ud,)

In order to compute the maximum fibre stress at
a critical section, we must include the direct stress
caused by the component of the reactions parallel to
the X axis in addition to the fibre stress caused by
bending. Therefore a direct stress diagram has been
drawn for typical gun carriage cradles.

For the case where the braking reaction it in the
rear from the front clip to the trunnion, we have,

A compression = uQt which is small and nay be
neglected as compared with the bending.

From the trunnion to the rear clip, we have at
the trunnions (just to right of trunnionslsee diagram]

a tension = X-uQ )

at the elevating arc contact section

a tension. = 2X-uQ1-Xc

Where the braking reaction is in front, we have from
the front clip to the "braking yoke" on the cradle
or ac a section through the point of application of the
tensions of the rods on the cradle, we have
a compression =uQ

From the braking yoke section the compression
increases to udt+B
at the trunnions

A compression =uQt+B(just to right of section)
A tension =2X-uQt-B

194

At the elevating arc section.:

A ^fltftfj.,to.<ri'»3

-ax-ua -B-X-

rfodffo s e-.- x c latUio'?

Shear diagrams for the 155 ra/m Filloux and 240 m/m

Schneider Howitzer show the variation of the shear in
these typical cradles.

To recapitulate if yt and yc = the distance to the
extreme fibres from the neutral axis at the trunnion
and elevating arc section, aod if At and Ac are tbej3£2yeo
areas of these respective sections, It and Ic cor-
responding moments of inertia, we have for the ex- nn9<j
treme fibre stress, ;i,n the critical sections of the /sib
ni si noJJsfesn fcniJUid erirf si^riw «»iso eriJ •;•

qiio Jno"jl sdJ fnoil i«9i
( ft = - ,., - + — - , ,for,toe braking rep A

action in the rear. _; gen

oJ Jeuj,/

iaa(x+xc+uda)-Bdb]y

or the braking re-

( *'

action in the front.

for the brak-
ing reaction
9w .Jnoi'l ai KI no ln rear

.AiftMO

[Q (x -xc+udo)]yc uQa

— —I — : - * + 8 ., , . 70

for the braking reaction

C AC iu f i.

in the front.

.-0 fi

iyoee »J(OH goi^sid «(< J

With Barbette and Naval mounts the gun recoils
in a cylindrical sleeve which forms part of the cradle,

195 aei

.

It is thus possible to conveniently locats the center
line of trunnions along the axis of bore or rather along
the line of action of the resistance to recoil which is
usually slightly below the axis of the bjpre. Therefore
the elevating reaction is simply due to the moment
of the weight ofowponent of" the recoiling mass about the
trunnions, though during the powder pressure couple we
have a momentary reaction depending upon the distance
of the center of gravity of the recqiling mass from the
axis of the bore to the powder pressure couple. We may
neglect this effect in battery as small and consider the
reactions out of battery as the maximum stress con-
dition. -

In the design of -a sleeve cradle it is highly de-
sirable to locate the .pulls symmetrical with the axis
of the bore, this being completely done by grouping
the two separate hydraulic and recuperator braking
systems equally distant and symmetrically above and be-
low the axis of the bore as in the 16.* model 1918 rail-
way mount. In smaller mounts the spring or recuperator
systems are symmetrically distributed about the axis
of the bore and the hydraulic braking is affected usually
in a single cylinder below the axis of the bore.

The hydraulic and spring or recuperator cylinders
are usually clamped in short distributed bearings to _______

the cradle proper. The reaction or thrust of the cy-
linders is taken upon shoulders at the end of these
bearing contacts. Hence a shoulder bearing surface may
be regarded as the point of application of the eccentric
load due to the pull reaction, on the cradle.

Two typical stress diagrams for barbette sleeve
cradles are shown for the 12" barbette mount, model
1917 and for the 16" railway mount, model 1918.

Considering the reactions on the cradle of the
.12" barbette carriage, model 1917 in figure (12) we
have, for the bending moment diagram, from the front
clip to the trunnion section, we have, a uniform increase
in the B. M. from uQd to -•

196

DIRECT STRESS

5HERR

BARBETTE
MODEL. OF I9H
CRRDUE.

Hg.

197

In passing from left to right of the trunnion section
there is no abrupt change in B. M. since the X re-
action being approximately on the longitudinal neutral
axis of the cradle is no longer an eccentric load.

The maximum bending moment occurs at the section
through the point of application of the eccentric
load due to the reaction of the hydraulic cylinder
against the cradle proper.

If Xj!, * the distance to this load d^ * the dis-
tance to the neutral axis, where Ph * the hydraulic brak-
ing, the bending moment at this section becomes,
to left of section, and

Pndn-Qt)ud +x'+xn)-2YxJ) to right of section.

In terms of forces to the right of this section the
B. M. becomes,

phdh~Qt(xi~xh~ud)+Ye(xe~xh)*xede to left of «ection»
and Qt(x^-xn-ud)-Ye(Xg-xJ)-Xede to right of section.

Since the elevating reaction is always very small
in this type of cradle, its influence on the B. M. direct
stress and shear is practically negligible .

Prom the direct stress diagram, we note a maximum
tension (sometines compression) between the trunnions
and the section through the point of application of the
eccentric load due to the hydraulic pall. In passing
to the right of this section the direct stress drops
in magnitude equal to the hydraulic pull.

The shear diagram shows no reversal of shear along
the cradle.

Considering the reactions on the cradle of the
16" railway howitzer, model 1918 (fig 13) we find a.
symmetrical distribution of the braking, the trunnions
being located practically along the axis of the bore,
thus reducing the elevating reaction to practically
the weight effect of the recoiling mass out of battery
and the neutral axis of the cradle through the
trunnions and along the axis of the bore. There

19!

}

n

1

4

«i j.

»- ,JW rr*

v^"

V

i—^f- 3~ i *

s

, . >»

uj,.

$

._ ^ ^ |_!

~t ~t -~-f-^-

I" — t

1 * *

4 4 1

I i >

r 1

„

c

> -* « *

fl p ^|

• 4 0 ^ 6 5f!

-•.

U

U

E^i

i

• bn*

O\RE1CT STRESS

4s ^c /••

— ^^

•

aoiloee

add 0^ I*ii?a r. ?> ol

•rose aai^Bib it-

^^^^ SHERR

_ r^
^j

! &n i 9 d

_

\NCH RRU-W/AY HOV/\TZE1R
MODE.U OF
CRADLE

Fig. N3

199

being no eccentric pull on the recoiling mass we
have a distributed load due to the weight of the
recoiling mass as shown in the figure

In the 8. M. diagram we find the B. M. at
the trunnions to be relatively small it being merely
equal to that due to a distributed load equal to the
weight of the recoiling mass, together with ths
friction caused by this loading. w

Thus, the intensity of loading =

x +x

*'Y3 "50 HTCH
Hence the B. N. at the trunnions, becomes,

t .
jtt Wr x Wr

+ u d

!R*<jirt $*u oi tx xoaebqaJ

Further between the trunnions and the application
of the braking to the cradle by the brake cylinders
thrusting against the shoulder bearings for the
cylinders on the cradle, we have a tension equal to
the total braking, thus causing a direct stress of
tension in addition to the bending.

To recapitulate, for barbette sleeve mounts, if
y= the distance to the extreme fibres from the neutral
axis at the section through the point of application
of the maximum eccentric pull due to the hydraulic
braking or at the trunnion section when the pulls are
symmetrically balanced, A = the area of the cross
section and I = the moment of inertia of the section,
we have for the extreme fibre stress

For barbette sleeves with eccentric pulls:—
Section at eccentric load -

• ', »x • leo
(Q. (ud+x *xv ) + 2Yxy, )y X— uQ

ft= — i ~ at left of section,

I •> ( AT o -

and

(P^-Q^ud+x'+XhVzYx^jy X-Ph-uQt
^t= " - i at right

of section.

200

?or barbette sleeves with pulls symmetrically
balanced about the axis of the bore:
Section at trunnion:

STRENGTH OP CYLINDERS AND The strength of a
RECUPERATOR PORGI>K3S. recuperator forging is

a matter of vital im-
portance since in
modern artillery the

tendency is to use higher and higher pressures con-
sistent with the various cylinder packings used,
and to stress the forging higher with smaller
factors of safety. Hand in hand with this goes the
metallurgical side where improvements in the quality
of the steel with higher ultimate strength are con-
stantly being made. High stresses in the recuperator
forgings as with high ultimate strength and low
factors of safety reduce the weight of the carriage
and its cost considerably . Weight of course is of
fundamental consideration for mobility. Hence it is of
importance to calculate the stresses in the cylinder
walls to a considerable degree of accuracy.

The maxim stress in a recuperator forging is a
combination of the following:

(1) A bending fibre stress normal to a
plane section perpendicular to the
longitudinal axis of the forging which

is caused by the external reactions exerted
on the forging during firing.

(2) A radial compression stress along a
radius of the cylinder or normal to

a cylindrical surface which is equal to
the pressure in the cylinder for the
inner surface.

201

(3) A tangential hoop tension, which is
normal to a plane passing through the
longitudinal axis of the cylinder.
Obviously these stresses are principle stresses
and with the aid of Poisson's ratio we may arrive at
the resultant intensity of stress. In a first ap-
proximation however it is sufficient to consider the
tangential hoop tension alone, the effect and magnitude
of the other stresses small.

Consider now a single cylinder 1" long, and sub-
ject to an internal pressure p, and external pres-
sure pt and of inside radius R0 and outside radius Rt
respectively.

Further let

r = the inside radius to any differential lamina

of the cylinder wall (in inches)
dr = the radial thickness of the lamina

pr = the radial compression at radius r (Ibs/sq.in)
pt « the tangential or hoop tension (Ibs/sq.in)

£ » the modulus of elasticity.

Cj = the longitudinal strain.

er 3 the radial strain

et = the tangential or hoop strain.

Then, for the equilibrium of a differential
lamina dr, of length "1" along the axis of the cylinder
and a peripheral length equal to the circumference,
we have,

2prlr-2(pr+dpr)l(r+dr)=2ptldr

hence

dPr d(rp,

-pp-*1 - — » E-

d,

Pt a ~Pr-r — - - -T-^ (D

V

Let us further assume plane transverse sections
to remain plane under pressure. This assumption is
reasonably close to actual conditions for plane trans-
verse sections some distance from closed ends, and in
the case of a recuperator forging where the intensity

202

of longitudinal stress, i. e. the bending stress on
transverse sections, is relatively small, except for
extreme fibres from the neutral axis of the transverse
section.

We are, therefore, not greatly in error in as-
suming the longitudinal strain to remain constant
over the entire cross section, hence,

vswcd c.:

ej = - (pi ) = constant^1 q°<

. ii ; edJ lo

where p^ = intensity of longitudinal stress,

— ev nt oi Jost

hence pt-pr=k (2) ^ MU(

dPr dP

Pt"Pr= ~2Pr"r cT^— or k + 2pr= -r ^7-

therefore dpr ^

k+2Pr " ~

Integrating, log(k+2pr)= - leg r* + c or k+2pr= — -

r

c k
hence pr = — - - - (3)

^ r* &

^c_ k «*•»«• qoc

2r* 2

,

Substituting (3) in (2) where pr=po

r =RQ inside conditions

r = Rt outside conditions

c k
hence P Q-

po 2

.

c k

P = ' - - oJ
1 2R2 2
i_

p _p _ _ fK\ -39 88T9V

ro ri vo^

,T a to

203

Now eliminating c and k, respectively, we find

t „

c - -- - - and k =

r i <i -3*1 )HoH I1 j3* ~"o3o S
Substituting in (5) we have,

— - - _ I _L^^^_J-_.-»_ T-. V_JT_«^«—

r RZ-R2 R2-B8 r2 (6) ( Apparent

( stress.

o .

p = , — + _^ — (7) (

HJ-B* Rt2-R« r» )b, Uiq»i«

The stress corresponding to the actual strain
produced in the material, which is the basis of stress
limitation imposed, (assuming m = 3 Poisson's ratio),

becomes,

.noiaoeJ qoo/i ecU n*rfJ **el «x*»i* •* ool»e»Tq»oc itJt-8T

Eet=St=E(^+~ - £i) (8) £i Actual

£ m£ raE

( stress

) corres-
p,. p. P] ( ponding

n o f f r * *\ /n\ ^891^8 «U-J ,

Eer=S_= -E(— + — + — ) (9) ) to actual

( strain

Where (6) and (7) are substituted in (8) ami (9) and

o2

i-i-ii. p °

F^-RQ

Jotet«<4 bn* { .^f'J)

the above expression reduces to Clavarine's formula.
Birnie's formula is a modification of the above
assuming p^ = 0, that is no external longitudinal
tension. Usually pi is relatively small and hence

•* A

(8) and (9) simplifying to, Uw 1<,01»J*<
eriJt lo lifiw co - ebiaJyo «dJ ^o *noiap9^ oocti sd*

204

Eet - S - - — + = -3-2 i- — (10)

* 3 R«-Rt 3 R»-Rt r»

rg 3 R.- R,

The maximum hoop tension, therefore becomes,
2R«+4R«

which gives slightly higher values than Lane's when
simplified, that is,

Rg * R;

St* P o«-R2 Lame's foraula (13)

Kt Ro

From the above formulae it is evident that the
radial compression is always less than the hoop tension.

With large bending fibre stress due to external
reactions on the recuperator forging,

pj is no longer equal to zero

The maximum stress which is the hoop tension becomes,

(14)
max

THICKNESS OF WALLS BETWEEN Considering two parallel
ADJACENT CYLINDERS. cylinders bored in one forg-

ing (fig. ) and passing
a longitudinal plane section
through the center lines or

axes of cylinders (1) and (2), we have either half of
the forging above or below this plane section in
equilibrium under the internal hydrostatic pressures
(which now are external with respect either half) and
the hooo tensions of the outside and common wall of the

205

two cylinders. Further if the two cylinders are
under pressure pt and p respectively and neglect-
ing the small variation of the hoop tension for dif-
ferent radius, we have for a close approximation

ptCT, + T, + «) - ptdt + ptdf

where Tt and Tf » thickness of cylinder walls (1)

and (2) respectively.
w » total width of common wall between the two

cylinders .
dt and df= the diameters of the respective

cylinders.
pt = the assumed allowable hoop tension fibre

stress.
Simplifying,

Pt

For a correction due to the fact that the hoop
tension is not constant but varies sligatly with
the radius, we nay augment w by decreasing pt to 0.9

Pf

Further due to symmetry

Pd P2da

and ^ " ~*

and substituting in the previous equation, we have

w =

1-8 Pt

which gives the minimum thickness of wall between
two cylinders under pressures pt and p8 respectively.
Evidently the maximum simultaneous pressures in the
two cylinders should be considered together.

206

ALLOWABLE STRESSES IN Though this matter will be
CYLINDER WALLS. taken up in detail in practical

design applications, certain
limitations could profitably
be mentioned here.

Cylinders should be tested for strength at pressures
considerably higher than would be used in service. It
is imperative that even under test pressure the elestic
limit is not exceeded. Test pressures should be at
least 1 — and preferably twice the maximum service

pressures and these test pressures should not exceed
3 the elastic limit of the material or 4_ proportional

limit.

TOP CARRIAGE The forces exerted on the

top carriage are the reactions
of the tipping parts and the
supporting forces of the plat-
form, or bottom carriage, or

ground and axle for a trail carriage. The reaction
exerted by the tipping parts on the top carriage may be
divided inter-
CD The trunnion reaction.
(2) The reaction of the elevating arc of
the tipping parts on the pinion of the
top carriage.

The tension of the equilibrator chain
or rod where an equilibrator is used.
These reactions are balanced by the supporting
forces exerted at the base of the carriage.

In figure (14) the reactions on the top carriage
are shown.

Considering now the reaction of the tipping parts
on the top carriage assuming that an equilibrator is
not used. Taking moments about the hinge point A(as in
previous discussions), we have, when the gun has recoiled
a distance X out of battery.

207

1

noiJBupe svcds sri-i ni ssui^v

i«J ,§•

-iR. 14-

.III TsJqsriO aaS fa - 2+S nie^l-^ aoo^ri woM

:ods 1H 'jo BT6

208

2X(ht-it sio 0)-2Y(lt cos 0+ht sin 0)

- E cos(ee- 0)[ht-j cos(9e-0)]-E sin(ee-0[lt-j sin(ee-9)]

'ta

Mo" (Kps+Wpx cos 0)

2X»Kr+Wt sin 0+ • cos 9(

J

(Krs+Wrx cos 0)
2Y=Wtcos 0- . sin 9e

and Krs+WrX cos 0

in _

J

Substituting these values io the above equation,

Krs+WrX cos 0

r+Wtsin 0+ ( • • )cos 9e](htcos 0-ltsin

J

Krs+WrX cos 0

-!Wtcos 6-( : ) sin 6e](lt+ cos 0+btsin

J

[KP+Wtsin 0+ ( , )cos 9e](htcos 0-ltsin 0)

0)
J

Kps+WrX cos 0

cos(ee~ 0)(ht-j cos(9e-0)l

-(Krs+WrX cos)sin(9e-0)[lt-j siD(

Expanding and simplifying, the above reduces to:

Kr(htcos 0-ltsin 0+S)-wtlt+»rX cos » Mta (1)

How htcos 0-ltsin 0+S = d See Chapter III.

where d is the moment arm of Kr about the hinge point A.

209

Further due to the displacement of the recoiling
parts a distance X from the battery position, the
center of gravity of the tipping parts is displaced
a distance ffrcos ^x from the initial trunnion

Wt
position.

The moment of the weight of the tipping parts
about A, is therefore,

r

Wt(lt--r-X cos 0) = *tlt-Wrx cos 0
™t

Hence from equation (1) we observe that the reaction on
the top carriage is equivalent to the total resistance

to recoil through the center of gravity of the recoiling
parts together with the weight of the tipping parts act-
ing at a distance IE x CQS ^ from the trunnions. There

fore the line of action, is equivalent in effect to
the resultant of the trunnion and elevating arc reaction.
This is almost obvious from first principles since the
reaction of the top carriage on the tipping carts must
balance the resultant of Kr and Wt: hence by the law of
action and reaction, the resultant reaction of the
tipping parts on the top carriage is therefore equal in
magnitude and direction to the resultant of Kr and Wt.

With a balancing gear we have in addition to the
trunnion and elevating arc reaction on the top car-
riage, (which now have different values from the pro-
ceeding) the tension of the equilibrator chain or rod.

By exactly a similar analysis as in the above, the
reaction on the top carriage reduces to the resultant

of Kr and Wt where the line of action of the component

•

Wt if now disolaced a horizontal distance __r_

^ X cos /) -

the distance which the center of gravity of the tipping
parts in battery is placed backwards from the trunnion
position, when the balancing gear is used.

210

In figure (14) is shown the various reactions on
the top carriage together with a force polygon. Thus
from the space diagrams of reactions obviously the

lines of action of the resultant of the trunnion re-

?,
action and elevating arc reaction intersect at a com-

mon point which necessarily lies along the line of
action of the resultant of Kr and Wt where the com-
ponent of Wt is displaced a horizontal distance

X cos <6 from the trunnion axis.

*

In the vector polygon of forces we note that by
vector addition, K+»t-X+T+E

Further for the equilibrium of the top carriage,
X+Y+E+Ha+Va+Vb = 0 hence K+Wt+Ha+Va+Vb = 0
The above results are exceedingly valuable in
graphical methods as will be outlined later for ob-
taining the various reactions throughout a gun car-

riage.

on ic-. ;pe «i .noijse 1o enrl edi eiol

•if- fioifmuiJ arfo
SUPPORTING REACTIONS Top carriages have been

OW VARIOUS TYPES OF q classified in Chapter I, into
TOP CARRIAGES. l[l) the ordinary type with

side frames and connected at front
or rear by cross beams or trans-

oms, which contain the pivot bearing, (2) pivot yoke type
used on small mobile mounts and (3) trail carriage.
The supporting reactions in the ordinary type of

top carriage are the H and V components of the pivot

-• v ct ri Kd/i H **> riiwi fi^£.f*f
bearing which is usually in the front and the V com-

ponent exerted by the traversing circular guides in the

rear. Sufficient horizontal play is allowed so that the

•
reaction of the horizontal traversing guides is only

•

vertical, the H component being taken up entirely at the
pivot bearing.

As a typical class (1) top carriage we may illus-
trate by the Vickers 8", Mark VII, British Howitzer.

; siU a-oil --SJ.TBCJ ni EJieq

211

Further let

1 = distance between supporting reactions measured

horizontally in the direction of the axis

of the bore at 0° traverse.
A = the front pivot point.

• r

B = the resultant of the distributed vertical re-
actions of the horizontal traversing arc guide.
1 k = the horizontal distance to trunnions from

B in the direction of the axis of the bore

o
at 0 traverse.

ht = height of trunnions above the traversing
guides.

S = the perpendicular distance from the trunnion
center to the line of action of the resistance
to recoil which necessarily passes through
e center of gravity of the recoiling mass.
= height of horizontal component of pivot re-
action above the horizontal traversing guides.
= weight of top carriage proper.
= moment arm of W about B.

Considering fig. (15) we have for the horizontal
component of the pivot reaction, Ha=K cos /5
and taking moments about fl, the center of pressure
of the traversing guides, we have,
Kd-W^-t*Wrx'cos 0-Wtcltc+Val-Haha = 0

( .

Wtlt+Wtcltc-Wrx cos 0-K(d-ha cos 0)
) hence Va= • ' '

*-ha cos 0)+Wt(l-lt)+lftc(l-ltc)-»-H'rx cos 0
( and Vb =

where for low angles of elevation, d=htcos 0+S-ltsin

d' = htcos ^-«-(l-lt)sin 0+S

212

Fig. 15

213

and for high angles of elevation,
d*ltsin 0-btcosl 0-S

d' = (l-lt) sin 0+htcos 0+S

ymn = the horizontal distance along the axle from
the center of the wheel bearing pressure.

Considering, max. traverse, right handed, at
max. elevation, the reactions on the axle to the
left of the section, become,

(1) The components of the trail con-
necting arm reaction on the axle:-
X,Y and Z together with a couple Mxy

in the horizontal plane.

(2) The vertical reaction exerted by the

left wheel, Sa. Therefore at section
"mn", we have,

(1) Bending in the vertical
plane:

(2) Bending in the hor-
izontal plane:

<in lbs->

(3) Shear in the vertical
plane :

X+Sa (Ibs)

(4) Shear in the hor-
izontal plane:

X (Ibs)

(5) Torsion about the y
axis, or in the x z
plane: T=X ZBn

(6) A direct thrust:

Y (Ibs)

Thus, we have bending in two planes combined with
torsion, and a direct thrust as well. Then for a

214

round section, as would^ Dually be the case, we h*»»,jon6

f 3 - - + - _ Max% normal fibre

°-78J thrust on outer layer

•oil »fx* etu c,ns.t8ifa 1*5 (Ibs/sq.in)

gniised Jaedw &dJ lo -iscfnso sr)J

, .-
Oj elxfi eri^ no

iis-U od^ io

-ralxs ori^ no noj ";s iniJoso

The m-axi/num fibre stress, therefore, becomes

_ ._an£tq Ifi-tnoxiiori erid ni

f =-f + /¥**"£ Jo* en ]80lj18v ftbs/sq.in)

3 4

• a 2 * j •

2

which should not exceed — of. the elastic limit of the
material to be used.

As a typical class o'f pivot yoke type, consider
the reactions on a Barbette or Pedestal mount, figure
(16) and a pivot yo.ke top carriage used on a trail
carriage, fig.ure (16A). In the first type, the lower
bearing sustains both horizontal and vertical com-
ponent reactions, whereas the upper is merely a
floating bearing and therefore sustaining only a hor-
izontal component and designed to prevent bending in
the lower pivot.

In the second type, the middle bearing has a
tapered fit within the axle, and therefore sustains
both horizontal and vertical components, but suffers
no bending moment since the axle is free to rotate.
To prevent the top carriage and mount from rotating about
the axle a lower cylindrical vertical pivot fits within
an equalizer bar below the. axle, the equalizer bar
being attached to the trails. (See Theory of Split
Trail - next section).

In this type of mount it is more convenient to
compute the supporting reactions in terms of the hor-
6 10^ nsriT .Ilsw 8£ Jsind.* joetifa £ b.~.

215

foul as o

l«3JUi»tr bn* isJ-nosx

aoioiq Sai<JBV8is 04 z^ibsi V.Q absa
-lev isdJ diiw jip*i 2oiJ*voI» no

REftCT\ONS ON
P\VOT >TOKE. TOP

cerfT

1 J II

^4^

IS 8OO

~n ale {-

aie

:®n — -ZH

216

izorual and vertical components of the trunnion.

If

ne = angle made by radius to elevating pinion
contact on elevating rack with the ver-
tical.

j = radius of elevating rack

Then for the horizontal and vertical components of
the trunnion reaction, we have,

Fe+Ks

>K cos 0+ ( ) cos ne ) In battery

j (

Fe+Ks )

<K sin 0+Wt-(— : ) sin ne (

J )

and

Ks+Wrb cos 0
2H =K cos 0+ ( ) cos ns ) Out of

J ( battery.

Ks+W_b cos 0
2V=K sin 0+Wt-( ) sin ne (

For the elevating gear reaction, we have

Fe + Ks

in battery

Ks+Wrb cos 0

j

out of battery

In the Barbette or Pedestal Mount, figure (16)
let,

xt = distance from center line of pivot to

center of trunnions.
yt = height of center of trunnion from bottom

of rivot.
T " radius of pivot bearing.

217

r*f = radius of floating bearing.

y^ = height between bottom of pivot and top

of floating bearing.
Then, Va = 2V + E sin ne

and 1 (

Hb= ~T ( 2Hyt+2V(xt-rp)+Etj+(xt-rp)sin ne-yt cos ne]
"m

and Ha=Hb+E cos ng-2H

In the pivot yoke trail top carriage, fig.(16A), let
x^ = distance from center line of pivot bo center

line of trunnions
yt = distance from center of axle to center of

trunnions .
ym = distance from center of axle to center of

equalizer beam.
Then, Va=2V+E sin ne

and H(j= - [2Hyt + 2Vxb+E(o+xtsin ne-ytcos ne)]

Ha=Hb+2H-E cos ne

THEORY OF SPLIT TRAIL. The object of a split is

primarily to give a large
aperture between trails for
the gun to recoil at maximum
elevation and maximum traverse

When split trails are used it is also desirable to dis
tribute the bearing load on spades when the gun shoots
at maximum elevation. T"his is accomplished by the

use of an equalizer bar connecting the two trails, or
more strictly the trail arms, beneath the axle, the
equalizer laying usually in a horizontal plane and
pivoted at its center by a vertical pin through the
center of the axle. With split trail and equalizer,
a pivot yoke type of top carriage s-hould be used.
The elements of a slip trail mechanism are :-
(1) The two trails with their spades
which are connected by a vertical pin

218

n o I i s v

5 7 ' &

2| *» KT^

I ^ x i...

/

toannoojifid
"^ "V i

i

n

219

to two trail arms or trail connect-
ing pieces, at either end of the axle.

(2) The trail arn or connecting pieces

are free to turn about the axle in a ver-
tical plane and are prevented from slid-
ing along the axle by thrust shoulders.
The moment about the axle of the trail
reaction is balanced by the moment
about the axle of the shear reaction
,,£t the equalizer bar.

(3) The equalizer bar is usually de-
signed to rotate in a horizontal
plane about a vertical pin through
the axle. Thus, with a split trail

we have the two trails, their connect-
ing pieces to the axle and equalizer
and the equalizer bar, connecting
the trail arms and pivoted about a ver-
tical pin which passes through the
center of the axle.
We have the following possible motions:-

(1) A free rotation in a horizontal
plane of either trail, about the
vertical pin in the trail arm.

(2) A constrained rotation in a vertical
plane about the axle of either spade,

the constraint being due to the equalizer
bar.

(3) A constrained rotation in a

horizontal plane of the equalizer bar
about a vertical pin through the axle.

MAXIMUM BLBVATION:

Let x and y = horizontal coordinates in longitudinal
and transverse directions respectively,

220

i. e. along and cross wise to the bore

at zero traverse.
Z - vertical coordinate.
AXAVAZ = the component reactions at the left

spade (positive direction towards

muzzle (Ibs)
8xByBz = the component reactions at the right

spade (Ibs)
Sa and Sb = normal vertical reactions for left

and right wheel respectively. (Ibs)
Qa and QJJ = shear reactions of equalizer on

trails A and B respectively (Ibs)
dj, = horizontal distance or span of equalizer

between trails which it connects (in)
de = vertical distance from center line of

equalizer to center line of axle through

wheel hub. (in)
£Mav = moments of the components of A about the

axle. (in Ibs)
SMQV = moments of the components of B about the

axle . (in Ibs)

jc_ = distance from spade to axle. (in)
yo = distance from ground to center line of axle.

(in)
2Ma[, = moments of reactions of A about vertical

pin in left trail am.
SMfj 3 noments of reaction of B about vertical pin

in right trail arm.
Taking moments about the center pin of the equalizer,

we have,

Qa 111 = Q. Ik.
a 2 b 2 hence Qa = Qb = Q. Therefore,

for moments about the axle, we have

-Q de = 0

hence, 2Ma * 2Mb

We have for unknowns,

Sa Sb

221

( Sight unknowns

Equations for solution:

ZX

£M

ZY= 0

ZMy =0

X = 0

= 0

2Mbh= 0

)

) Nine solutions

(

)

(

)

We therefore would expect either 2Mah or ZM^j, not zero

This is physically justified since on extreme traverse
one of the wheels and trails must be in contact. This
is met constructively by usually introducing a show
attached to the trail which comes in contact with the
wheel upon traversing.

If N = normal reaction between shoe and wheel,

dD = perpendicular distance from vertical pin

on trail arm to vertical plane through wheel,
i. e. to line of action of N.
We have for maximum traverse in a right banded rotation,

^bh=^ ^n *nus introducing an additional unknown
N. The solution is, therefore, statically possible
either introducing the above equation or omitting it
entirely.

METHOD OF SOLUTION

Assume maximum traverse right handed turn,
Let 0ffl = maximum angle of elevation.
8m = maximum angle of traverse.

222

Ks = resistance to recoil at maximum elevation

,,. . :v*ri eW

(Ibs)

Xy = horizontal distance to projection of center
of gravity of recoiling parts measured from
base line AB. (in)

yg = 0 assumed approx.

Zg = height of center of gravity in battery above

ground, (in)
g = vertical distance from ground line to center

of pressure on spade . (in)
Wg = weight of total system, gun + carriage. (Ibs)

g

= horizontal distance from AB to W

,,

Li *7

y A'B1 = distance between vertical trail pins.
Then, the resolved component of Ks through the center
of gravity of the recoiling parts, become,

/<MS
Kscos 0m cos em, along the x axis )

Kscos 6^ sin Q , along the y axis )

( r. i ri T

K sin <L along the Z axis )

xp = distance from AB to either vertical trail pin.

yab = distance between A and B, trails completely
, neswJed tv i-ioa = n 11

,
spread.

Taking moments about A B, we have,

: muaix
hence, W 1 -Kscos 0 cos 9m(Z.+ g)+Kssin

• x. - -,

.pa «vo4s art* r- .:is

and Az+ Bz= V Ks sin ^«-(Sa+ Sb ) (2)

Next take moments in a horizontal plane about the left
spade, and we have,

. .t £ v a 1 s J^ab

=

.

em. xg-Kscos Bf.cos 9m -j- = 0

223

K cos 0m

hence Bx = - (0.5 yab cos 9n~Xg sin 6m) (3)
yab

Further Ax+Bx=K3cos J0m cos 0m

hencs AX=KS cos Am cos 9m-Bx (Ibs) (4)

For moments about the vertical pin in trail arm for
left trail, we have, Ayx_-Ax 0.5 y/^'B1 ~ 0

0.5 AxyAnj»
hencs Au = - (Ibs) (5')

XP
Now if we take moments about the axle, we have ZMgy^

Az *0-W*> = Bzxo-Bx^o^>
therefors (A-B ) (Z+g)

'Z "Z X,

but

lience Az =~ 2x ~~^~ (6)

Bz=W>»+Kssin e)m-(Sa+Sb) - Az (7)

Let

X, Y and Z = components of the reaction of the

trail arm on the axle,
MXV = moment reaction of trail arm on axle, in

the X Y plane.

Considering moments on the left trail and trail
arm together about the axle, we have,

AzxQ-Ax(Z0+g)-Q de = 0 hence, the horizontal

shear reaction of the equalizer becomes,

*.

Next consider the various reactions on the trail arm,
and we have,

224

= ° along the x axis,
-Y+A = 0 along the y axis.
-Z+AZ =» 0 along the z axis.

and further, -Mxy+Ay(xo-xp) '0 In the x y plane
Therefore, the reactions of trail arm on the axle,

becomes, A_xo-Ax(Z0+*)

X=A+A » z o * o + A., (Ibs) (9)

do

Y = Ay (Ibs) (10)

Z « A, (Ibs) (11)

Mxy» Ay(x0-xp) (in. Ibs) (12)

Of AXLS MAXIMUM TRAVBR8B, MAXIMUM ELEVATION:

This critical section of an axle is at a section
near the center where the axle becomes enlarged for
holding the vertical pivot of the top carriage. If
the axle is made straight, we have no torsion on the
section but mersly bending in a vertical and
horizontal plane. If, however, the axle is underhung
for clearance and lowering the top carriage, in
addition to the bending, we have torsion as well, the
nagnitude of the torsion depending upon the depth of
the underhang.

Let mn be the section under consideration near
the center of the axle.

xnn ^mn an^ zmn = ^9 component distances from
the center of contact of the
trail connecting arm and
axle.

REACTION BETWEEN RECOILING During the counter re-
PARTS AND MOUNT IN COUNTER coil, we may distinguish
RECOIL. between the accelerating

and retardation period so far as the. reactions between

225

the recoiling parts and mount are concerned. The re-
actions during the acceleration however are of exactly
the same character as during the recoil only of less
magnitude. Therefore, from either the point of view
of the internal reactions or stability of the mount,
we are not concerned with the acceleration period of
counter recoil.

Therefore let us consider the various recoiling
parts and mount coming into play during the retardation
period of counter recoil, -

Let (see figure 18)

xt and vt = coordinates, along and normal to
bore, of front clip reaction with
respect to center of gravity of
recoiling parts.

x and y = coordinates, along and normal to
bore, of rear clip reaction with.
respect to center of gravity of
recoiling parts.
Qt s front clip reaction.
Q2 = rear clip reaction.
Wr = weight of recoiling parts.
0 = unbalanced retarding force exclusive of
f rict ion.

^0 - distance from center of gravity of re-
coiling parts to line of action of 0.

n = coefficient of friction = 0.15 usually.
d1 = distance from front wheel ground contact

to line parallel to tore through center

of gravity of recoiling parts.
lr= horizontal distance from line of action

of Wr to front wheel ground contact.
x = displacement along bore or guides from

out of battery position.
Ma = moment of reaction of the recoiling parts

on mount about front wheel contact and

ground.
Then, for the motion of the recoiling parts, we hava.

226

REACTION BETWEEN RECOILING PRRT5
RND MOUNT IN COUNTER RECOIL

RECOILING PRRT5

227

d x

0+n(Qt+Q2)+Wrsin 0=-mr — - (1)

d t

Qt-Q2=Wrcos 0 (2)

and 0 d0-ftlx1-Qfx8+n Qtyt-n Q8y2 = 0 (3)

Next, considering the reactions on the mount and taking
moments, about the point of contact of the front wheels
with ground A , we have,

(4)

- - d'tan 0+x J=MrA'
0

Substituting Eq. (3) and (2) in Eq. 4, we have
immediately

d"x

rQ )d +Wrsin 0.d -W_l =

* 2

or

(0+n(Qt+Q2)+Wrsin 0)d '-Wrlr=MA

Further from equation (1) (-mr ^ )d'- Wrlr= My

Q t»

Hence, the reaction on the mount during counter recoil
is equivalent to the total resistance to recoil acting
in a line parallel to the axis of the bore and through
the center of gravity of the recoiling parts, together
with a component in line and equal to the weight of the
recoiling parts.

If further, we let,

Fy = the recuperator reaction.

RQ = total guide friction
RS+P 3 total packing friction.

Bx = total counter recoil buffer reaction.
Then 0 = B '+£„.. n - F,,

aa ) = n Wr Cos 0 (approx.)
and the overturning force, passing through the center
of gravity of the recoiling parts and parallel to the

228

bore, becomes,

i dv
- [Fv-Wr(sin 0 +n cos 0)-Rs+p-Bx)= -rar v — - (Ibs)

GRAPHICAL CONSTRUCTION AND Very often it is more
EVALUATION OF THE REACTIONS convenient to evaluate
IN A GUN CARRIAGE. the various reactions

by graphical methods.
Graphical constructions

are of special value since they give a vivid picture
of the relative magnitude of the various reactions.
Further the method is comparatively simple and the
closing of force and space polygons combined with
overall methods gives an admirable check. The ac-
curacy of the method even -with rough layouts is suf-
ficient for the computation of the various reactions
required.

If we consider the kinetic equilibrium of any
piece of the carriage, we have, by introducing the .
kinetic reactions or inertia forces with the actual
reactions exerted on the piece, a dynamic problem re-
duced to a problem of statics.

For equilibrium of the piece, we have,

2X = 0 )

ZY = 0 ( for a coplanor set of forces.
ZM = 0 )

Now the considerations ZX = 0, ZY = 0 are met
by the vector diagram of reactions having a zero re-
sultant, that is the vector polygon of the piece
closing.

The condition ZM = 0, requires a consideration
of the lines of action of the forces in a space
diagram in addition. Since the moments may be taken
about any point, there can be no resultant moment exist-
ing. The condition 2X = 0, ZY = 0 implies the result-

229

ant force to be zero, but does not imply the
existence of a couple. Condition ZM 3 0.

indies that a resultant couple cannot exist.

A graphical method, therefore, always consists of
two sets of diagrams,

(1) a space diagram of forces and

(2) a vector diagram of forces.

The space diagram requires a layout proportional to the
actual piece under consideration and the placing on
this diagram the lines of action of the forces. The
force diagram requires a layout proportional to the
direction and magnitude of the various reactions
exerted on the piece. The two diagrams must be
carried on simultaneously since the direction of a
resultant required in a sp^ace diagram, is obtained by
the vector addition of the forces which are the com-
ponents of the resultant. Since Vector addition is
commutative, the order of Vector addition is immaterial.

REACTIONS ON THE RBCOILIMG PARTS

The known reactions consist:

(1) The powder force along the axis of
the bore Pfc . (IJbs)

(2) The inertia force along an axis parallel

to the bore and through the center of

d •
gravity of the recoiling parts - - mr x's

Zt r ^

(3) The weight of the recoiling parts
acting vertically through the -center of

gravity of the recoiling parts - - Wr.
The unknown reactions consist:

A a; ** ^ ** •*• u £ " *

(1) The resultant braking force B

Ibs.

(2) The front and rear clip reactions
Q, and Q2 Ibs.

230

The lines of -actions of these forces however
are known or can be readily determined.

Procedure:

Layout a space drawing proportional to the
dimensions of the recoiling parts, showing the
assumed lines of actions of the various forces.

See fig. (19;. 2

Since P^ - »r — — = K the total resistance

to recoil which is
assumed as known,
we have the effect of PK and m ^ * equivalent to,

b G "r

dt2

(1) a couple Pfc %

(2) a force K through the center of
gravity of the recoiling parts parallel
to the axis of the bore.

Since a couple and a single force may always be
reduced to an equivalent single force, we have (1)
and (2) combined into a single force K acting at a
distance above the axis through the center of
gravity of the recoiling parts equal to

P. eh

p 9 (in) ( where CK is in inches)
K

The reactions 0 and Q due to the friction in

12 i

the cradle sleeve make an angle u = tan -*f with the

normal to the guides, where f = coefficient of friction

Q

= 0.15 usually. Hence u = 8.5 approximately.
Referring now to the force polygon or diagram, lay
off K in the direction and equal to the magnitude of
the total resistance to recoil.

Lay off K = a b

From b lay off b c = WR, the weight of the re-
coiling, in magnitude and direction.

Draw K + Hf = a c

231

232

Referring now .to tha space diagram lay off K at a
perpendicular distance °b9b

K

above the center of gravity of the recoiling parts
and parallel to the axis of the bore. At the intersect-
ion of K and Wp , draw J k parallel to a c until it
intersects the line of action of the motion of the
reaction Q2 at k.

From c of the force polygon, lay off c d and fn
the direction of the rear clip reaction QZ.

Draw k c from k to the intersection of Qt and 8
in the space diagram.

Draw a d parallel to k c in the force diagram
until it intersects a d at d. This limits and de-
termines the magnitude of 0 in the force diagram.

From d, draw d e parallel to B and a e parallel
to Q . The intersection of a e and d e determines B
and Qt respectively. Thus from a combination of the
space and force diagram we obtain Q2 B and Qt respective-

iy-

REACTIONS ON THE CRADLB.

Referring to figure (20):
The known reactions consist:-

(1) The rear clip reaction Q2 (Ibs)

(2) The front clip reaction Qt (Ibs)

(3) The weight of the cradle Wc (Ibs)

(4) The braking force B (Ibs)
The unknown reactions consist:-

(1) The trunnion reaction T (Ibs)

(2) The elevating gear reaction E (Ibs)
The direction of the latter being' known.

Referring now to the force diagram lay off a b =
in the direction and proportional to the magnitude of
flf. From fa draw c parallel and equal to B the brak-
ing force.

Draw a c.

Referring now to the space diagram J k from the

233

in the force polygon, to the intersection of Q± •

In the force diagram, draw c d = Q and
parallel to Q.t . draw ad.

In the space diagram draw J L parallel to a d
to the intersection of Wc.

In the force polygon draw lfc equal and parallel
to Wc the weight of the cradle. Draw a c.

In the space diagram 1 m parallel to a e to the
intersection with E at m.

From m draw m n to the trunnion axis, which gives
the line of action of the trunnion reaction T.

In the force polygon draw e f in the direction
of E and a F in the direction of T. The intersection
at f determines the magnitude of E the elevating gear
reaction and T tha trunnion reaction.

t» yf "X

REACTIONS OH THE TIPPIHG PASTS.

Locate the trunnions along the resultant of the
"battery position of Wr and Wc --- See upper diagram.

Without balancing gear:-

Considering the external forces on
tipping parts, we have, the known reactions,

(1) The total resistance to recoil K (l"bs)

(2) The weight of the tipping parts Wt
(Its)

The unknown reactions,

(1) The elevating gear reaction E (l"bs)

(2) The trunnion reaction T (l"bs)
the direction of E "being known.

In the space diagram lay off X parallel to the
bore and at a perpendicular distance from the center
of gravity of the recoiling farts = pbe (. .

In the force diagram, lay off ab = K and be = Wt.

Draw ac.

In the space diagram, lay off J k from the inter-
section of K and Wt parallel to ac of the force diagram

234

235

I

<L
6

U

o

o
t-

0 -o

&

5

\

u
o

i

of

ft:
a:
o

0.

o

K

cu

V)

236

a:
Q

O

d

I

0

U

237

£
<c

s

cr

uJ
u
ot
o
u.

I

Q

LJ

o

d

0.
CO

u a

(3

m

£

K

ID
N

<t

DO

0

iZ

Q

*^'pfe; k '

y

3

^* ^

- nl/f r x

b

i

/ 7 '{ 8^

\

£

/ A §5

\
\

i

0

c

/ \/\ i g

/y ' t

/7 ! s

\
\

\
\
\
\

f /

\

x/ /

\

I

/ / /

i

' / /

*/ '

'//

^-X'/

/ / "

/ // /7 /

Aw !'

Its

nr^

/ A

^ > \ 1 Z o

^^-"'

I r>-

238

and t o the intersection of E.

Draw k 1 to the trunnion axis in the space diagram.
The line of action of T is then along h 1 produced.
In the force polygon, draw cd parallel to E and a d
parallel to k 1 of the space diagram. Their inter-
section at d determines the magnitude of £ and T
respectively.

With balancing gear:-

Determi nation of the balancing gear reaction
R. On the space diagram lay off Wj. the weight of the
tipping parts in its battery position as well as the
line of section of R. From the intersection of R and
Wt draw o m. This must be the direction of the result-
ant of Wt and F since the condition is that we have
no moment about the trunnions when W^ is in its battery
position.

In the diagram below, lay off W. and R and draw o.m
parallel to o m in the space diagram. This determines
the magnitude of the balancing gear reaction R.

Referring to the force diagram, lay off a b equal
and parallel to K the total resistance to recoil, and
be = Wt the weight of the tipping parts.

Draw ac.

In the space diagram K is at a perpendicular dis-

D p

tance _fc_- from the center of gravity of the recoil-
ing parts and V^ at a distance

Wrx cos 0
— * from its battery position, where

x is the displacement in the recoil

At the intersection of K and Wt draw j k parallel to ac
to the intersection of R.

In the force polygon draw cd parallel and equal to
F. Draw a d.

In the space diagram draw 1 k parallel to a d of

the force polygon to the intersection of the line of
action of E ac 1. Draw 1 m to the trunnion axis, thus
determining the line of action of the trunnion re-
action T.

In the force polygon draw d e parallel to m E and

239

a e parallel to 1 m, thus determining the magnitude
of E and T respectively,

R3ACTION3 ON THB TOP CARBIAGB

Without balancing gear:-

The known reactions consist:

(1) The weight of the top carriage
Htc (Its)

(2) The trunnion reaction T (Ibs)

(3) The elevating gear reaction E
(Ibs)

The unknown reactions consist:

(1) The horizontal component of the
pintle reaction - H (Ibs)

(2) The vertical component of the
pintle reaction N (Ibs)

(3) The front vertical clip reaction M
(Ibs)

The lines of actions of these forces are given
from the construction of the piece.

Referring to the force polygon fig. (25), draw ab
=T equal to the magnitude and in the direction of T
the trunnion reaction. Draw be parallel and equal to
E the elevating gear reaction.

Draw ac. In the space diagram draw j k parallel
to T.

At the intersection of j k and E produced draw k 1
parallel to a c in the space diagram to the intersection
of Wtc.

In the force polygon draw c d equal and parallel
to Wtc- Draw a d Prom 1 in the space diagram 1 IE
parallel to a d to the intersection of N produced.

From m draw m n to the intersection of H M. Draw
a e in the force polygon parallel to mn in the space
diagram.

We thns have d e in the force polygon = N and
ef = M and ja= H.

240

Thus the pintle reactions H and N and the clip
reaction are determined in magnitude and direction.

With balancing gear:-

The "known reactions consist:

(1) The weight of the top carriage Wtc

(2) The trunnion reaction T

(3) The elevating gear reaction E

(4) The balancing gear reaction B
The unknown reactions consist:

(1) The horizontal component of the
pintle reaction H

(2) The vertical component of the pintle
reaction N

(3) The front vertical clip reaction M
The lines of actions of these forces are given

from the construction.

Referring now to the force polygon fig. (24) Lay
off ab = T and be = E. Draw ac.

In the space diagram from the intersection of
T the trunnion reaction and E elevating reaction pro-
duced at K.

Draw k 1 parallel to ac of the force polygon. Con-
tinue in the force polygon c d = R the balancing gear
reaction. Draw ad. In the space diagram draw in parallel
to ad and the intersect! on of W^ at m. In the force
polygon draw de.

Draw ae .

In the space diagram draw mn parallel to ae to the
intersection of N. Fron N draw n o to the intersection
of M and H. In the force polygon draw a f parallel
to o n. From E in the force polygon draw e f parallel
to N to the intersection of e f .

Draw f g and & a as shown.

Thus we determine the reactions M, N, and H
respectively.

241

REACTIONS ON THE ASSEMBLED CARRIAGE GUN ASP CARRIAGE
TOGETHER.

Location of the weight of the total mount:-

Assuming a static reaction of 200 Ibs. under
the spade, we lay off o'm = 200 Ibs.

Then o N = Wg = 200 under the wheel contact.
The resultant of o'm and o n » W3 obtained by the
additional construction lines o'q and op. Hence we
determine from the triangle of forces the line of
action of *L. The external reactions on the as-

9

sembled carriage consists of :-
The known reactions -

(1) K = the total resistance to re-

coil.

(2) Ws = the weight of the total mount.

The unknown reactions -

(1) The horizontal spade reaction Ha.

(2) The vertical spade reaction Va.

(3) The normal reaction under the wheels

The direction of these forces are known.

Referring to the force polygon lay off ab = K the
total resistance to racoii and be = weight of total
system W3.

Draw ac.

In the space polygon from the intersection of

< and Wg draw j k to the intersection of the reaction

Va-

Prom k draw k 1 to the intersection of Ha, Vb

at 1.

Referring to the force diagram draw ad parallel
to 1 k of the space diagram to t"hs intersection of
c e produced.

We thus determine c d - Va, d e = Vb and e a - Ha<

242

Thus the reactions Ha, Ba and V^ are determined
in magnitude.

PROCEDURE IN THE CALCULATIONS FOR THE PRINCIPLE RE-
A.CTIONS IN A GUN CARRIAGE MOUNT.

(Illustrated by calculations on 240 n/m Hewitmer)
REOUIRBD DATA.

Type of Gun Howitzer

Diameter of bore d (in) 9.45
Total Weight of recoiling parts Wr(lba) 15790

Weight of Powder Charge W (Ibs) 40

Muzzle Velocity v (ft/sec,) 1700

Travel of Shot in Bore u (in) 160

maximum 60°

Angle of Elevation 0 ninimum 10°

short 3.74

Length of Recoil b (ft) long 3>g0

Intensity of Powder Pressure p^dbs/sq.in) 32000

Initial Air Volume of Recuperator Vai 2970

(cu.in)

Initial Air Pressure of Recuperator Pai 576

(Ibs/sq . in )

243

INTKRTOR BALLISTICS.

Maximum Powder Pressure on Breech 2,245,000
F » Pb = 0.7854 d2pm (Ibs)

Maximum Powder Pressure on Base of 2,005,000
Projectile pm (Ibs)

P°=A: (Us)

Mean Constant Powder Pressure jQ * *7°Q

5.36 x 160

5*36U 1,350,000

1 = twice abscissa of Max. Pressure

- ~ —)* - 1 3.996

POD = Muzzle Pressure on base of breech

622,000 Ibs.

Vsl. of free recoil: 7f

wVm + 4700 W

= 50.25 ft/sec

Wr

Vel. of free recoil - Shot leaving
Muzzle

0.5W Vm

wr

40.50 ft/sec,

Time of Shot to Muzzle

t s — — i- 0.01175 sec.

1 2 12V,,

244

Time of Expansion of Free Gases
-

ob

32.2

0.01538 sec,

Free Movement of Gun while shot
travels to Muzzle

•_ u"(w+0.5W)

l~ 12(Wr+w + w)

0.31 ft.

Free Movement of gun during Pow
der Expansion
P v t*

0.7179 ft.

Total free Movement of gun; Pow-
der Pressure Period:
I • Z4 + X,

1.0279 ft.

Time of Powder Pressure Period
r - tt * tf

0.02713 sec,

BRAKING PULLS AMD STRESSES IN CYLINDERS.

x axis taken along bore: v axis taken normal to bore.

Mass of Recoiling parts

»r
"r '' 32.16

15790
32.16

491

Constant of Stability
C » 0.85 to 0.9

Calculations only
for max. elov.

Height of center of gravity of
recoiling parts above ground
h (ft)

Calculations only
for max. elev.

Stability Slope

elf.

Calculations only
for max. elev.

245

Total Resistance to

Max.Elev.
Recoil

491 * 50.75

Hor.Elev.

,2

K =

2(b-£+VfT)

(Ibs)

2 (3. 74-1. 0279+50. 75*. 02713)
152,000

Variable Resistance to re- Calculations only for max.
coil in battery (at elev.
horizontal elev.)

jnV

K--

rf

2[b-E+VfT- - -
2 M,

(Ibs)

Variable Resistance to Re- Calculations only for max.
coil out of battery (at elev.
horizontal elev. )

k = K-m(b-E+ - )
2m.,

Initial Recuperator Re-
action, Pai = approx.
1.3Wr(sin 0m+0.15cos0m)
Ibs. (unless given)

1.3 x 15790 (gin 60+0. 15cos60)
= 19300 used
18800 Ibs.

Total Initial Recuper-
ator Pull, Pai = P^i

100 d.

(Ibs)

18800+100x2.938

19094

da = diam. of recuperator

rod. (in)

0.

Effective Area of Recuper- 35.756
ator Piston -
Aa (sq.in)

Initial Air Pressure

ai

(Ibs/sq.in)

18800
32.6

576

246

Initial Air Volume Vai (cu.in) 2970

Final Air Volume Vaf(cu.in)
vaf - Vai -12 Ab

a

Final Air Pressure
P., "".'"

Final Recuperator reaction

2970 - 12 x 32.6 x
3.74 = 1510

S76

1214

1214 x 32.6 = 39600

af

= p

af a

air

Paf= approx. 2Pai(lbs)metallic

J_Distance from axis of bore to 3.038+ 3.850
mean guide contact r(in)

3
3.4444

Distance between clips 1 (in) 86.25

J. Distance from axis of bore to 16.365
center line of hydraulic pis-
ton e^ (in)

J_Distance from axis of bore to

line of action, of recuperator 15.656

reaction ea(in)

Assumed coefficient of guide
friction u = 0.15 to 0.25

Guide friction constant

2u

» Af

l-2ur

0.15

0.15

86.25 -2x0.15x3.44
.00352

247

Total hydraulic Pull 152000+157908in60-18800( 1+.0635)
(max. elcv.) 1+.0663

» 137500

UAfeh (Ibs)

Total hydraulic Pres- 2 hydraulic cylinders:
^-100 dn ^ '

diam. of brake rod

Ph»P^-100 dn ^ 68750-4.72x100= 68280

(in)

Effective Area of Hy- 31.2
draulic Piston

Max. Pressure in Hy- 68280

- = 2200
draulic Cylindsr 31.2

' (Ibs/sq.in;

Inside Diam. of brake 7.874
cylinder

dih- 1,

(in)

(dn* diam. recuperator
rod)

Outside diam. of brake 9.450
cylinder

Hoop tension in brake

cylinder wall 2200(== — ) = 12150

9-45' -7.875'

Ibs/sq.in.

248

Max. pressure in recuperator 1214
cylinder v

Paf=?ai

Ubs/sq.in)

Inside Diam. of recuperator 7.087

cylinder

dia = 1.13/A0+0.785d| (in)

Outside Diara. of recuperator 8.267

cylinder

doa (in)

Hoop Tension in Recuperator
Cylinder Wall

,d§a+d!a

1 d 2 -d * '
uoa uia

. in)

.8.267+7.087
12140==1) =8020

.267-7.087

Inside Diam. of compressed
air storage tank d (in)

84-66

Outside diam. of compressed
Air Storage tank doc ^in^

Hoop Tension in compressed
air storage tank

Paf (

9.45

a — — __2

1214 C—^— * ' — )
9.45-8.466'

11000

d2 -d2.
oc ic

(ibs/sq.in)

Width of Wall between ad-
jacent cylinders^ (in)

Hoop tension between adjacent - - - - -
cylinders

p = phdih*Pafdia

1.8 w

.i n)

249

GUIDE, ELEVATING GEAR AND TRUNNION REACTIONS:

'.

x axis taken along bore: v axis taken normal to

bore.

Coordinates from center of Xs 37.843
gravity of recoiling parts
to front guide reaction yt*-3.038
xt and yt (in)

Coordinates from center of xg » 48,4O7
gravity of recoiling parts to
rear guide reaction ya= 3.86

x, and y2(in)

J_ distance from center of 16,365
gravity of recoiling parts to
brake piston rod axis e^

J_ distance from center of 15,656
gravity of recoiling parts to
recuperator piston rod axis
ea (in)

•U^esa-r -:-™* U ( $* o

Max. powder reaction P^T (Its) 2,245,000
(See Interior Ballistics)

J_ distance from center of 6.13
gravity of recoiling parts to
axis of bore e (in)

Front guide reaction: gun re-
coiling in sleeve:

Fe+Pneh+Pa-Wrcos0(x2-uy2)

Q 5S I -

0.15 to 0.2 (Ibs)

250

Rear guide reaction: gun recoil-
ing in sleeve

Q =•

Fe+P£en+Paea+Wrcos

(Ibs)
Front guide reaction: gun recoil- 2,245,000x5.13+137500

ing in guide below axis of bore 37. 84+48. 41-0. 15><6. 91
Fe+P£en+Paea-Wrcos0(x8-uy8)

t + x -u(yt+y

Rear guide reaction: gun recoil-
ing in guides below axis of bore

Q

xl6. 365+19094x15. 66

-7895x47.63

2,245,000x5.13+137500
37.84+48.41-0-15x6.91

x!6. 365+19094x15. 66

+7895x37.38

162600

Max. guide friction
Kg » u(0t+a8)- (Ibs)
u = 0.15 (approx. )

Weight of Tipping Parts Wt(lbs)

Max. Resistance to recoil (dur-
ing powder period)

2Peu

Bg = 0.15(154800+
162600)= 47,620

21,021

=137500 + 19094+47, 620-
13670=191000

=152000+
5.13x0.15

2x2,245,OQQx
85.21

=192000

251

I distance from trunnion axis 3.73
to line parallel to axis of
bore through center of gravity
of recoiling parts s (in)

Radius to pitch circle of 35.57
elevating arc. j (in)

Angle between "y" axis and the 60°
radius to elevating pinion con-
tact with elevating arc 9e *
0 + ne

Elevating gear reaction (in
battery) E - Fe]K'a. (Ibs)
Angle of E with horizontal

Top carriage trunnion reaction
(in battery with balancing
gear)
2X=K+Wr3in0+Bcos99+Rsin9r (Ibs)

2Y=Htcos 0+Esinee-Rcos9r (Ibs)
(E is sans with or without
balancing gear)

Top Carriage Trunnion reaction Not used.

(out of battery with balancing

gear)

2X=K+Rsin9r+B'cos9e+Wtsin 0 (Ibs)

2Y=Wtcos0+E'sin9e-Rcos9ii (Ibs)

(E1 is same with or without

balancing gear)

11,513,884*191,000x3.73
35.57

» 344,000
Not used.

Estimated Weight of Bocker
W- (Ibs)

Neglected as small

252

Horizontal Distance from Trunnion
to center of gravity of rocker

hr (in)

"measured to rear"

Not used.

Angle between line of action of
rocker reaction on cradle and
"y" axis. B

I distance from trunnion to
elevating sere* or normal to
rocker cradle contact,
k = x^os B+y^sin B (in)
"xm and ym coordinates of
rocker contact with cradle
from trunnion to rear and
down" .

+ 30

29. 43x. 866+15. 71
*0.5 - 33.35

Rocker Reaction on Cradle
M= -^— — (Ibs)

344000^35.57
33 . 35

367000

Elevating Bear Reaction
(out of battery)

E1 =

Ks+Wrbcos 0

(Ibs)

Calculations max,
elev. in battery,

Top carriage trunnion reaction
(in batterv)(X and Y components)
2X=K'+lft3in 0+E cos 6e (Ibs)
0-E sin 6

2X=197000+18200+
17200=381200
2Y=10510-198,000=
-287,500

253

Top carriage Trunnion Reaction
(out of battery) (X and Y com-
ponents )

2X=K+Wr sin 0+ E cos 9e (Ibs)

cos 0-Ein

(Ibs)

Calculation at max.
elevation in battery,

Vith balancing Gear: Distance
from trunnion to center of
gravity of tipping parts (in
battery) along x axis:
xt (in)

Not -used.

Radius of bell crank
(balancing gear)
r,, (in)

Not used.

Balancing Gear Reaction:
2Wtxt cos 0

ra(l+cos

(Ibs)

Not used.

(very approx.)
or calculated from layout
0m = max. elevation.

Angle made by balancing
gear: reaction with "y"
axis

9,

(See layout)

Not used.

Rocker Trunnion reaction

(X and Y components)

2Xr =M sin B-E cos 8e~V*

sin 0 (Ibs)

2Yr= E sin 9g -¥* cos B

(Ibs)

183500 - 172000
11500 = 2Xr
297000 - 318000
- 21000 = 2Yr

254

Total shear reaction of trunnion 190600 + 5750
on cradle, - X'=X+Xr (Ibs) 196350 = X1

Y'=Y+Yr (Ibs) -143750-10500'

154250 = Y1

Total spring reaction of Top
Carriage on trunnion

sin

(Ibs)

cos 0 (Ibs)

10000 * .866
8660 = X3
10000 * o.S =
5000 - Ye

Total rigid bearing reaction of
top carriage trunnion
Xb = X -Xs (Ibs)
Yb = Y -Ys (Ibs)

190600 - 8660
181940 = Xfc
-143750-5000=
-149750= Y,

Bending moment at cradle section 8660 x 5.5 +
of trunnion 181940 * 2.9 + 5750

Mx= Xsa + Xbb + Xrc (Ibs) *.0.9 ~ 580780
My = Ysa + Ybb + Yrc (Ibs; 5000 x 5.5 - 149750

*2.9 - 10500 x 0.9 =
-416,950

Resultant B. W. at cradle section
of trunnion

(in Ibs) /580,7802 + 416,960'

716,000

Max. fibre stress due to bend-
ing

a 10.18 M (Ibs/sq.in)

10.18 x 716000
—8*

355

n

rJ
evi

7.5

3.6 —

TfcVJNN\QN PIN

~ n -.

256

SHEAR REACTION OF CRADLE.
ON TRUMNtON PW :

RERCT\ON OF ROCKER ON
TRUNNION P\N :

RERCT\ON OF TOP CRRR\RG£.
ON TRUNNION

\9O6OO
RERCT\ON ON P\N \N X PURNE..

257

258

CALCULATIONS FOR STRENGTH OP CARRIAQ1 AXLE

Proposed 75 m/n St.Chamond

50° Elevation and

22- traverse:

Maximum Peak Resistance to Recoil - - assumed

at 20,000 Ibs.

The resistance to recoil may then be divided into
a horizontal and vertical component in the vertical
plane of traverse. T"hen, the horizontal component in
the vertical traversed plane, nay "be divided into a
component along the horizontal axis of the mount and a
transverse component at right angles to the longitudinal
axis of the mount.

The components in the vertical traversed plane
are:-

Horizontal comp. = 20,000 * cos 50° = 12820 Ibs.
Vertical comp. = 20,000 * siii 50* = 15320 Ibs.

The longitudinal and transverse "horizontal com-
ponents are:-

Horizontal comp. = 12820 * cos 22.5° = 11800 Its
Transverse comp. = 12820 * sin 22.5° = 4900 Its.

859

Then, 15320 + 4000 = 19320 (Total Downward Force)

S x 130 = 4000 x 120.25 + 15320 * 128.2 - 11820 * 47.2
4000 x 120.25 = 481000
15320 x 128.2 = 1970000

11820 x 47.2 =

19320

14550

4770

2451000

558000
1893000

S = 14,550

4,770

A, + B2 = 4770
»nef

260

12800 x cos *2 = 11800

12800 x Sin22- * 4900

2MA » 8X x 142.4 -11800 x 71.2 + 4900 x 128.2 » 0

11800 x 71.2 = 840000

4900 x 128.2- 629000

211000

H800 ... B • 2110°° • 1481 )

1481 L42-4 (

A, = 10319

£M about vertical pin for loft trail
Ay 126.38 * 10319 x 55.2

4900 .'. A, 4500

4500

400 By - 400

IM axle - A, 130 -10319 x 32 * Ba 130 - 1481
130C/1, -8Z) = 10319 x 32 - 1481 x 32
=283000

.'. AZ-BZ = 2180

Bz= 4770

2AZ - 6950 .*. Az = 3475 )

8Z = 1296 j

£M about left wheel base in Z Y plane:

-4900 x 41.2 + 15300 x 30 + 4000 x 30 - 4900 x 6

+ 3475 x 41.2 - 12.95 x 101.2 - Sp x 60 » 0

- 4900 « 41.2 * - 202000

49000 x 6 » - 29400

-1295 x 101.2 = - 131100

- 362500

261

15300 x 30 * 459000 722000

4000 x 30 = 120000 362500

3475 x 41.2=143000 359500
722000

Sg = 5980 )

SA = 8570 )

Reactions on Trail Axle.

X and Y reaction on vertical pin of left trail:

Ex = 10319 t Ey = 4500

B. ¥. in XY plane on axle:

Ey x 10 = 4500 x 10 = 45000 " # XY plane:

Thrust along X axis = 10319 ± Shear reaction of equal-
izing bar.

Thrust along Y axis = 4500

Thrust along Z axis = 3475

Shear reaction of Equalizer bar =
452000 - 331000

7.75
Thrust along K axis

15600

10319
15600

25919

EXTERNAL FORCE.S ON RXUE
FOR SECTION -(m-n

&

-45OOO

n

262

Section n-n 5" x 5"

Torsion = 25919 x 2.2 = 57000 (" *)

(B.M.zy) = 3475 « 12.2 + 8570 x 2'6

42300

22300Q B. M. jjTy = 265300 " *
265300
(B.M.zy) = 25919 x 12.2 - 45000

316000

45000 B. M. = 271000 (" *)
~27100(T

f

I \ * 30032- •= Ci > :'0---. * 01 * ^1
265300 x 2.

. 12700

5 x 25

271000 x 6 13000
fy ' 125 " 25700

n 54 625 n
32 32

. 46500
•1.4

12850 + S\ x 257002 + 46502
12850 + 13620 = 26,470

BICAPITDLATION Qf gQ R HUT. AE OH THR TNT^BKAT.
BBACTIOHS THROliaHnilT A GHH C*RBT«GB.

F = Powder reaction (Ibs)

B = Total braking force not including

guide friction (Ibs)

263

"b = distance from center of gravity of recoiling parts
to line of action of 8. (in)

R = total guids friction (Its)

r = mean distance from center of gravity of recoiling
parts to guide friction (in)

e - distance from center of gravity of recoiling parts
to line of "bore. (in)

Pn= total oil pressure on the "hydraulic piston. (Ibs)

P'= the hydraulic reaction plus the joint frictions
^ (stuffing box at pistons) (Ibs)

Pa= the total elastic reaction (due to compressed air
or springs) (Ibs)

Pa* the total elastic reaction plus the joint frictions

(Ibs)
Cj, 3 distance from center of gravity of recoiling

parts to line of action of Pn. (in)

ea = distance from center of gravity of recoiling

parts to line of action of Pa. (in)

d}, = stuffing box or rod diam. of hydraulic cylinder.

"V5 (in)

da - stuffing box or rod diam. of air cylinder. (in)
Q = normal front guide reaction (Ibs)

0 = normal rear guide reaction. (l"bs)
xt and yt = coordinates from center of gravity of re-
coiling ]!>arts to front guide reaction, (in)

1 = distance between line of action of Qt and Qf (in)
x and yf = coordinates from center of gravity of re-
coiling parts to rear guide reaction, (in)

!fr = weight of recoiling parts. (Ibs)

0 = angle elevation.

u = coefficient of friction.

X. and Y = component trunnion reactions (Ibs)

Xr and Yr = component roc"ker reactions at the

trunnion (Ibs)

& = elevating gear reaction

J = radius to pitch circle of elevating arc. (in)
9a= angle between "y" axis and the radius to elevating
pinion contact with the elevating arc.

264

K * total resistance to recoil. (Ibs)

s » distance from center of gravity of recoiling parts

to trunnion axis measured along the "y" axis, (in)

Total resistance to recoil on recoiling vass. becomes.

K * B + R - Wr sin 0 (Ibs)

but B = Pn + Pa

where Ph = Ph + 100 dn ) assu>ing 100 1T)8. pCr

and i i (in diaro. for frictions

P. » P.+ 100 d.

) in stuffing box.

hence

K * P * ? + R - Vi sin 0

QUIDK OR CLI? »KACTIQ«8 TO QUIDS FRICTIOM.

Gun recoiling in sleeve, front guide reaction,

Fe+Bb-W_ cos 0(x. -uy. )

Qt - - - - ' - *— (Ibs)

xt+xf+u(yt-yt )

and rear guide reaction.

•

Fe-»-BbCWr cos 0 (x±* uy,. ) /1V .

\ (Ibs)

Gun recoiling in guides below the axis of the bore.
front guide reaction,

Pe+Bb-W.cos 0 (x -uy )
Q = - - - - - 2_ (Ibs)

and rear guide reaction,

Fe+Bb+Wr cos 0(x -uy )

Q^ = - - - - - V * (Ibs)

«,+ x4-u(yf+ya)

If R * xt+xt+u(yt-ya ) for sleeve guides

M = xt + xa-u(yi+y2 ) for guide below axis of bore
and

265

H * x -x +H (yt+yt ) for sleeve guides

* * xt-xt+u(yt-yt) for guides "below axis of bore.

then the total guide friction equals,

2(Fe+Bb)+W cos 0 . N
R = - - - u ,(lbt)

and for the total braking force B4

(K+W_ sin 0)M-(2Fe+W_ cos fS N) u
B = - 1 - 1 - (It.)

X +2 u b

In terns of tbe pulls, we bave for the clip re-
actions,

Fe + £Pa + 2Pe~lf cos ^(x~u)

Q , - — — (Ibs)
*t**t*u<*t-y«>

Fe+IPaea+ 2Pv«h+WP cos 0(x +uy )

" - i - J- (Ibs)

xt+
and tbe guide friction becomes,

2Fe+22P^eh+ 22Paea+ Wr cos 0 K
R = - (Ibs)
M

and tbe hydraulic pull in terns of the total re-
sistance to recoil and recuperator reaction, becomes,

M(K-£Pa~Wr sin 0)-u(2Fe+22P'efl+ N *fr cos 0) ,

« A r - A. a . ., r_ , _< (Ibs)

For approximate calculations, the guide friction
equals, 2u8dr

R " ~

From tbe foregoing analysis we observe, that tbe
guide friction and bearing pressures are reduced:

• (1) By increasing ths distance between
the clips.

266

(2) By balancing the pulls about the
center of gravity of recoiling

parts or bringing the resultant pull
closer to the center of gravity of
the recoiling parts.

(3) By "bringing the resultant friction
line of the guides closer to the
center of gravity of the recoiling
parts .

(4) By reducing the powder pressure
couple Fe, that is by reducing the
distance from the center of gravity
of the recoiling parts to the center
line of bore. The distance from center
of gravity of the recoiling mass to
the center line of bore should never
exceed 1.5 inches unless a friction
disk is introduced with angular notion
about the trunnion.

Stress QB

Let

Wc = weight of piston and rod or the weight of
recoiling cylinder. (Ibs)

d- = distance from center of recoil pull to section

"mn" adjacent gun of the gun lug. (in)
Imn = moment .of inertia of section. (in)'

y - distance to extreme fibre from -neutral

axis. Cin)

fnn3 nax. fibre stress (Ibs/sq.in)

then, W'

[B+ -2. (F-B)]dgy

-n

(Ibs/sq.in)

Trunnion and Elevating gear reaction:

When the gun is in battery the tipping parts
are balanced about the trunnion axis. This condition

267

implies that with the gun in battery, the center of
gravity of the tipping parts passes through the trunnion
axis. When the recoil is limited to a short movement
under the breech when the gun is fired at high elevations
the center of gravity of the tipping parts is placed
forward if the trunnion axis and the balancing gear or
counterpoise is introduced, balancing the weight of
the tipping parts about the trunnion. The trunnion
reactions are modified by the introduction of a
balancing gear.

Trunnion and elevating gear reactions when no
balancing gear is used:

(a) During the' acceleration period,

,Fe+Ks,
2X=K+Wt sin 0 * (— J ) cos 9 (ibs)

(ibs)

(b) During the retardation period,

Ks+W_x cos 0
2X=K+Wtsin 0+( • )cos 9e (Ibs)

Ks+W x cos 0

2Y=Wtcos 0-( ) sin 9P (ibs)

J

Ks+W_ X cos 0

E * (Ibs)

J

where x = the recoil displacement out of battery.

Rocker Reactions:

T^s reactions on the rocker are primarily three:

(1) The reaction of the trunnion upon
the rocker, Xr and Yr .

(2) The reaction of the elevating gear,
E.

268

(3) The reaction of the cradle , M,

and the weight of the rocker, 1fr.
If k = the perpendicular distance fro« the trunnions
to line of action of M.

B = the angle between the line of action of M and

the "y" axis.
h'r= the horizontal distance to the center of gravity

of the rocker from the trunnion.
J = the perpendicular distance from the trunnion
axis to the line of action if the elevating
gear reaction (i. e. equals the radius of
the circular elevating rack on the rocker).
Then, the cradle reaction on rocker, becomes,

Ej-Wrhr Fe+Ks-Wrhr

M = = (in battery) (Ibs)

k k

Ks-Wrx cos 0-Wrhr

= (out of battery)(lbs)

k

K»
approximately M = —

k
The rocker trunnion reactions become,

2Xr = M sin B -W^ sin tf-E cos 6e (Ibs)

2Yr =E sin 9e -Wr cos 0- M cos B (Ibs)

Layout of Balancing Gear:

Two types of balancing gear have been used ex-
tensively in gun carriage construction:

(1) A cam with chain type for small field
mounts .

(2) A direct acting balancing gear.
For type (1), let

Wt = weight of tipping parts. (Ibs)
hj. = horizontal distance from the trunnions to
the center of gravity of the tipping parts

269

(gun in battery) (in)
ro = equivalent radius of can at horizontal

elevation (in)

rn = final equivalent radius of the cam where the
cam arc has turned through the maximum
angle of elevation = 0 (in)
R = niean radius of can. (in)
dn= deflectioa of spring at zero elevation (in)
dQ= deflection of spring at maxim-urn elevation (in)
c = spring constant.

0= angle of elevation expressed in radius.
If ds - deflection of spring at solid height, take
dn » (J to j)d solid )

(
d =(^ t° i) d solid )

then _ *tht
"

rodn rndo

and dn-d0=(

To layout the radii of cam, we have 0 divided into

n parts, then,
tht

Wh cos

rtnt

r =

c(dn-r0A0)
Wh cos 0

cos 0

With a balancing gear of this type, the trunnion
reactions are modified and now become,

270

if T = the tension in the chain

d = the angle T oalces with the axis X(taken along
the axis of the bore)

Ks+Wrx cos 0+Fs

2X=K+"Vfc sin 0 + ( - • - ) cos 8Q - T cos d (Ibs)

J

Ks+¥_x cos 0+Fs
= 2Y = ¥tcos0-( jsin 6e + T sin d (Ibs)

J

The elevating gear reaction obviously remains as before
that is,

Ks+\frx cos
E = - £ - - - (Ibs)

for type (2), 1st

Tft= weight of tipping parts (Ibs)
ht = horizontal distance from the trunnions to
the center of gravity of the tipping parts
(gun in battery) (in)
x^ and yt = coordinates along and normal to bore

from trunnion to canter of gravity
of tipping parts (gun in battery)

0 - angle of elevation.

£5m = max. elevation

r = radius from tbe trunnion to the

crank pin which connects the tipping parts
to the piston rod of the oscillating
cylinder, (in)

R = reaction exerted by the balancing gear along
the piston rod of the oscillating cylinder.
(Ibs)

dt= moment am of H about trunnion (in)

d^= deflection of spring at horizontal elevation

d]j= deflection of spring at maximum elevation

(in)
c = spring constant

Hj = initial balancing gear reaction (0° elev.)
Rt = final balancing gear reaction (0° elev)

371

S = stroke of piston in oscillating cylinder (in)
pt = final air pressure in pneumatic balancing

cylinder (Ibs/sq.in)
p^ = initial air pressure in pneumatic balancing cylinder,

(Ibs/sq. in)

A = effective area of balancing piston (sq.in)
Vo = initial air volume (cu.in)

With a metallic balancing gear, the dimension of
the spring" may be approximated by the solution of the
following equations:

= cos 0_ ) from nrhich we may obiain d0, dv,

Q,
( s and c of the spring.

a S

r(l+ cos — ) ^
2 )

S = 2 r sin -

With a pneumatic balancing gear, we have, for a pre-
liminary approximation,

Pf

2Wtxt

S

( Pl ^ ( )

(

_ — •
r ( 1 +c<

£lDJi

\ ) (.CM . in /

)

3S2

1

(

Pi

)

S = 2r sin

*•

f. . Pf
(in; — = cos 0.

(

2

Pi

(approx)

)

With a direct acting balancing gear, the trunnion re
actions are modified and become,

Z72

if

R = balancing gear reaction (Ibs)
qr- angle between R and y axis

dt = moment arm of R about the trunnion at any

elevation 0 (in)

when the recoiling parts are in battery:

2X«K+Wtsin 0+E cos 8e + R sin 9r (Ibs)
2Y=Hft cos J0+E sin 9e -R cos 6r (Ibs)

Wtxt cos 0 21*txt cos ^
R » : = a (Ibs)

Ks + P^e

when the recoiling parts are out of battery :-

2X»K+R sin 6r+E cos 6e+Wt sin 0 (Ibs) )

(

2Y»Wt ces U + E sin 6ft-R cos er (Ibs) )

(
2Wtxfc cos t

R = . (roughly) (Ibs) (

r(l+cos— -) )

2 (

Ks+W_ x cos 0

Is (Ibs) (

J )

It is evident that th« elevating gear reaction
remains the same with or without a "balancing gear
while the trunnion r«actions are modified both by
the position and Magnitude ef the balancing reaction.

273

Strength of the trunnions

The critical section ef the truT>',i«»s is usually
where the trunnion joins the cradle. L«t, "«n* represent
this section. [See fig. (9)].

a = distance fro» "mn11 to center of top carriage

bearing .

b * distance from "mn" to center of rocker "bearing
MX = the bending moment at "mn" in the plane of

the X component reactions.
My= the bending moment at "mn" in the plane ef the

Y, component reactions.

M « the resultant tending moment on section "an".
f = aax. fibre stress (Ibe/sq . i'ff)

D = distance ef fhe trumnien at section "mn"
then

Mx= X.a+Xrb (in Ibs) and M =
My= Y*+Yrb (in. Ibs)

hence

/10.18 M
D = / (in)

Stresses in cradle or recuperator forging:
Let

Ql and QZ = the front and rear normal clip re-
actions .

xt and xg ~ the x" coordinates of these re-
actions with respect to the
trunnions.

dx and da = the distance of the friction co»-
ponents ef Q± and QZ from the
neutral axis .

B = the resultant of the braking pulls re-
acting on the cradle.
d-= the distance from the neutral axis to "B".

274

It = moment of inertia of a cross section at

the trunnions.
yt = distance of extreme fibre from neutral axis

at trunnion section.
ft= fibre stress due to bending and direct pull

or thrust at the trunnion section.
Ic = nonent of inertia of a cross section at
the point of contact of the elevating
arc with cradle.

AC = area of cross section, at the point of con-
tact of elevating arc with cradle,
y = distance to extreme fibre from neutral axis

of elevating arc section.
fc = fibre stress due to bending and direct pull

or thrust at the elevating arc section.
A^ = area of a cross section at the trunnion,
then

)yt UQ-

+ — • — for the braking reaction

in the rear,

ft = — - — i— + — i for the braking

^ reaction in the
front .

*x,

* x, • "" ~^«*~ •" w j^ • UVA

for the brak-
ing reaction
in the rear.

U°2

f = " ' " = ^^ + — i- for the bralc-

T A

Ac c ing reaction

in the front.

- -Jfc'^SJ APPENDIX-----
APPENDIX CHAPTER IV- INTERNAL REACTIONS.

BKACTIOHS AMD STRESSES IKDPCSD IK ELEVATING AMD TR«VERS-
IH8 MECHANISMS:

STRESSES DUE TO The reaction exerted on the
FIRING. elevating mechanism due to

firing equals,

In Battery, Out of Battery

Fe + Xs ?%v Ks*Wrx cos J0

J cos 20 J cos 20

where

F = max. powder force

K = Total resistance to recoil

YT = weight of recoiling parts,
r

x = displacement out of battery.

J = radius to pitch line of elevating arc fron

center of trunnions,
e = J_ distance fron axleof core to center of gravity

of recoiling parts.
S a J_ distance from line parallel to axis of

gun through center of gravity of recoil-
ing parts to center of trunnions.
It is highly desirable to reduce the reaction
E, since it stresses the teeth of the elevating
mechanism. To reduce this, we may,

(1) decrease "e" by so distributing the
mass of the recoiling parts as to bring
its masses ss near coincident with the
axis of the bore as possible.

(2) decrease "S" by bringing the trunnion
axis along a line through the center of
gravity of the recoiling parts and
parallel to the axis of the bore.

(3) increase "J" whenever feasible in
a construction layout.

275

276

In certain types of oounts as those contain-
ing a recoiling cylinder, the piston and rods "being
fixed to cradle, the center of gravity of the
recoiling parts is necessarily considerahly lowered
froa the axis of the bore and therefore "e " is in-
herently large. With large mounts, counterweights or
bob weights are sonetines introduced to decrease "e".
In this type of mount without a counterweight or "bob
•eight a friction clutch or hand brake are often in-
troduced on the elevating gear shaft or adjacent gear
shaft. Then E becomes limited to that required to
overcome the friction of the clutch or brake and a
large reaction on the elevating mechanism is thus re-
duced.

With a cone clutch, we have,

uPr

E = : — - , where P= total spring load.

r = mean radius of clutch
re = pitch radius of gear or

pinion,
n - coefficient of friction =

0.15 approx.
2« = cone angle

With a dislc clutch, we have,

), where P = total spring load.

r2= outer radius: rt= inner

radius of dislc.
k = total no. of friction

surfaces .
n = coefficient of friction

- 0.15 approx.

FRICTION OF TRUNNIONS Tn elevating, or traversing
AND TRAVERSING PIVOTS. a gun, a large amount of the

energy needed is that required
to overcone the friction of
the pivot about which the gun is traversed.

277

Trunnion friction:

During the elevating process the load on trunnions
equals the weight of the tipping parts, when the
trunnion is sufficiently free from binding, the con-
tact is along a narrow strip.

Then u

t
nR sin 0+R cos 9 = -r— ) u= coefficient of

where tan 0 * n lO friction.

Wt ) R = »or»*l pr«s-

. * .R (sin 0 tan £J+ cos 0)=~r — ( SUP«

) r = rsdius of trwa-
( nion.

and the friction moment

Mt= R tan 0 .r = — r sin 0

m

Since 0 is small, tan 0 = sin t approx.
hence W Wt

M+ = n — —r = 0.15 — •— r approx.
$o l 2 2

In starting n nay be as great as 0.25 an& proper
allowance should be nade.

Since the load brought on the trunnions
during firing is greatly in excess of that on
elevating the gun, the bearing contact may be
divided, one part to carry the major of the
firing load and the other to carry merely the
weight of the tipping parts. This is ac-
complished constructively by allowing play in
the bearing which sustains the firing load, and
holding the tipping parts for elevating or
transportation merely on a spring cushion,
the reaction of the spring, for a deflection
just sufficient to lift the tipping parts just
clear from the firing bearing, being equal to
the weight of the tipping parts. Thus it is
possible to reduce the friction by using a

278

smaller trunnion diameter, in tliat part of the bearing
that is spring borne since the bearing surface for a

nominal bearing pressure can be greatly reduced.
Pivot friction in traversing:

This friction will vary considerably according
to the type of bearing used. We will consider three
types of pivots, 1* flat circular pivot, 2° flat
hollow circular pivot, and 3° conical pivot. To esti-
mate the load brought on the pivot, let,

Va = pivot reaction or load (vertical )

V^ = normal load of traversing guides (vertical)

Wt = weight of tipping parts.

1 = horiiontal distance between Va and Vv

d U

l+= horizontal distance from W+ to Vv

v u U

Wc = weight of top carriage.

lc = horizontal distance from ¥t to V^

then ff 1 +W 1

Va = - load on pivot during traversing.

If Kt= the friction couple exerted at the pivot
during the process of traversing we have for the various
types of bearings,

1° for flat circular pivot:

The friction on an elementary zone = 2 re r dr

The moment of this friction about the center = — j2nr*dr

L 27a ro 2 2Vanro
The total friction = — <g — n/ r dr = —

ro o 3

Therefore for a flat circular pivot, letting n -
0.15,

Mt » 0.1 Va r0

279

2° for flat hollow circular pivot

The total friction evidently becomes,
— I * r'dr =

2Van r 2Van r'-r'

3
hence, letting n = 0.15

3°for conical pivot:

The intensity of vertical pressure on the projected
area of the bearing = y

n(rf-r*> V

If the cone makes an angle 2«, and pn equals the in-

tensity of normal pressure, then,

rd6dr rd9dr

the normal pressure on area — — = p_ — .

since B fJin «t

the vertical component of this pressure =
rdedr

rd 6dr

sina

but the pressure on the projected area rd6dr = p rd9dr
hence p¥ = Pn = ?.

2nrdr

the friction on a differential zone = n

2

the total friction moment, therefore becomes,

If then we let n = 0.15

280

VELOCITY RATIOS OF ELEVATING Elevating and travers-
AKD TRAVERSING MECHANISMS ing mechanism consists

usually of a train of
spur, bevel, helical
screw and worn gears.
l°)-Velocity Ratio of spur gear:

Since *trt = v»tra ) * = angular velocity

^ r = radius to pitch line.

we nave -± = — L - _L ( n = no. of teeth.

2°)-Velocity Ratio of Bevel gears:

Again w r = w r where r and r are the outside

1 t Z 2 1 2

radii of the gears:

The angle of coning for the first gear, equals,

r.

" *

tan 6t «— ( 6t = 1 angle of cone, )

or the second gear

tan 62 = — (62 = - angle of cone)

hence w

1 2

1 = tan 9. and - = tan 0
w

Therefore we may take any two common radii in ob-
taining tlie velocity ratios, again

3°)-Helical screw gears:

Tfe have for the velocity of the common normal,

w r cos

r2cos 9,
r cos 9

but also,

Pn =
then

= pcos

cos 9.

Pn

cos 9.

281

) 0t = angle be-
l*(F* tween axis
) of gear fl
( and perpend-
icular to

( common normal.

. ) 98 = angle between
( axis of gear
) #2 and per-
( pendicular to
) common normal.

( pn = common normal

) pitch.

( pt = circuraferent-

) ial pitch gear

( #1.

) n = circumferent-

tial pitch gear #2.

n = no. of teeth gear fl.

n2 = no. of teeth gear f2,

Hence — = — =

r cos9

r cos9
i 1

If 0 = the total angle between the axis of the gears

in mesh, then
since p = p cos Q = p cos 9

cos 9t =

cos e = "• si"

~fP2~2 PtP2cos

therefore

282

Further the axial pitches, become,

TBX = pt cot 6t and m2 = pa cot 9f

4°)"Velocitv Ratio Worm gears:

Though a worm gear is a specified type of nelical
screw gear when Si = 90°. it is convenient to consider
this type as a separate classification.

When 0 = 90°,

cos 6. *

= sin 9

therefore the axial pitch of one equals the cir-
cumferential pitch of the other.

The worm of a worm gear has one to two or three
threads while the gear has many threads.

Now, for a single thread worm,

rwsin 9
r., cos9

- — tan 9

Directly, we have.

%

P^T

but

2n

= tan9.w»r

wlw

; *g= ang. velocity
( of gear wheel .

) ww- ang. vel. of
( worm wheel.

) rg - pitch radius

( gear.

) rw= pitch radius

( of worm

) p = axial pitch of

( worm

) 9 = angle of helix.

— = — tan 9

Thus the ratio of angular velocities depends upon
the angle of the helix of the worm.
With a "n " threaded worm,

: nwP —
£n

g r
and — = n -^

ww rg

tan 6

283

p
In terms of the number of teeth, since = tan 9

w

p nw

n... ian 9 - —

2nr ng

and for a single threaded worm, since nw = 1

* -

Velocity ratio in gear trains:

Combining the previous equations from one pair
of elements to the adjacent pair, we finally arrive
at the velocity ratios of the first and last wheels
of the trains in terms of the number of teeth or radii
of pitch circles: In this combination, it is always
preferable to set the general equation up in terms of
the number of teeth rather than the radii of pitch
circles, for then the relations are independent of
the type of gearing and velocity ratios between a
meshing pair are inversely as the number of teeth or
threads.

Thus assume worm #1 to drive worm gear #2, while
bevel gear #3 on same shaft as gear #2, drives
bevel gear #4, then helical screw gear #5 on same
shaft as gear #4, drives helical screw gear #6 and
finally gear #7 on gear shaft #6, drives pinion #8.

Since 2 and 3, 4 and 5, and 6 and 7 are on same
shafts, we have,

then,

wwwwnn nn

13 5794 88

hence — * — * — * — = — * — * — * —

284

wt nt n4 n n
therefore — » — x — x — x —
w n n n n

8 1 3 5 7

If Tt =» torque on worm shaft #1 and T the torque
on pinion shaft and e the efficiency of
the total gearing, then T§wa = e T^

hence -p n xn xn Mn

Tt * — ( ) where Tt = required power

torque and
T$s load torque at

end of train.

REACTION BETWEEN GEAR PAIRS:- The efficiency of
EFFICIENCY. spur and bevel gears

is hifh compared with
helical screw gearing,
especially of the worn

gear type. The very large force and velocity ratio
attainable by the latter makes this type preferable.

1° Spur Gears:

For approximate calculations, the normal reaction
between 'the teeth will be taken at an angle of 20* with
the tangent to the pitch circles. The effect of
friction between the teeth is to cause the resultant
reaction to make an angle of 25° with the tangent to
the pitch circles.

Therefore if T is the torque to be transmitted,
the reaction between the teeth R, becomes,

T x 12

where T is measured in (Ib.ft)

r cos 25° , . .

r is neasured in (in.)

Ifhen smoother running is required with high
velocity ratios helical spur gears have been extensive-
ly introduced. If B = the angle between the normal
to a tooth surface and the tangent to the circumference
(i. e. normal to axis of rotation), then

285

T * 12
r cos 25 cos B

If b = tooth rim breadth, the mean pressure is dis-
tributed along a linear element - b sec 6 and

therefore the pressure on an element becomes
per linear inch, proportional to

T * 12

r cos 25. b the same as in ordinary spar
gearing.

2* Bevel Gears:

The reaction between bevel gears takes place at
the intersection of the common pitch circles of the
cone elements of the gears, and this intersection is
in the plane of the axis of the gearing. The neutral
reaction between the teeth makes an angle approximately
equal to 20° with the normal to this plane due to the
contour of the tooth. The tangential component pro-
duces no axial thrust. The component parallel to the
plane = P tan 20, where P is the tangential component.
This component is also perpendicular to the common
intersecting line of the two cones. If the cone angle
of gear #1 equals 28 then the cone angle of gear #2 =

The axial thrust for gear fl becomes, P tan 20° sin6
The axial thrust for gear *2 becomes, P tan 20° cos6
Further the radial reaction "between the teeth and there-
fore the radial bearing loads for gear #1 and gear #2,
becomes,

R' = /p2 4. (p tan 20° cos8)2 = P /l + (tan 20° cose)'

Where T x 12

P = ; and 28 - the cone angle of gear #1

r

71

2(- - 8)= the cone angle of
gear #2.

286

3° Helical Screw Gears:

Assuming the axis of the gears to make an oblique
angle £J t the angle 6 between the contact line of the
teeth and axis of gear #1 is given by the expression

P2sin 0
cos 9t =

while the angle 9 between the contact line of the
teeth and axis of gear #2, is given by the expression

p sin 0
cos 9. = ,. . * ..

where pt and pz are the respective circumferential
pitches of the two gears.

The reaction between the teeth makes a resultant

angle i with the normal to the contact line, where
tan i = n the coefficient of friction

Then, if Tt is the external torque exerted on
gear #1, we have T± = R cos (9t - i).r±
while if Tg is the torque on gear #2, Ta=R cos(92+i).r.
Work expended = T w

w ' r cose
Work delivered = T8wt
Then the efficiency E becomes,

T w cos(9 +i)cos9
E = -2-2- ? -

tt cos(0t-i)cos92
The reaction on the teeth is given by

Tt
r.cos(9 -i)

T sin(6 -i)
R sin(et-i)= ^

287

and the thrust along gear shaft f2, is

t
R sin(8 + i ) = —

r cos(9t-i)

The total radial bearing load for shaft of gear #1
balances,

T cos(6 -i) T
R cos(9 -i)= -^ _— i-— = -1
rt cos@f-i; rx

and the total bearing load of gear shaft #2
balances, ^

R. cos(9 +i)= — -

4° Worm Gear:

Though, worm gearing is a special case of 3°, a
separate analysis will be made due to the greater use
of this type of gearing as compared with helical
gearing when the shafts are not at right angles.

Let xx and yy* be the coordinate axis along
and perpendicular to the axis of the worn in the
plane perpendicular to the radius of the pitch line of
the worm through the common pitch point as origin.

Let S = the angle that the contour of the tooth
makes with the normal to the xy plane at the pitch
point, and 6 = the angle of helix.

Let R = normal component between worm and gear
tooth,

nR » friction component between worm and gear

tooth,
then the axial thrust along worjn wheel is

X - R cos S cos 9 - nR sin 6 and the turning
component on the worm is

Y = R cos S* sin 6 + nR cos 0

and the thrust tending to separate the teeth is
Z = R sin S .

It is to be noted that tan S = tan S cos 6

288

If T_3 torque applied to worm gear

Tg=» torque on gear wheel
then,

Tw=Yrw and Tg=Xr^ rw= radius of worm gear

rg= radius of gear wheel
To determine the efficiency, .vs hava

but — = — tan 0

then „!

cos S cos0-n sin8 .

e = — — — — — — — — — — tan 6

cos S'sin0+n cos6

n tan0

cosS1

-) tan 0

n
cosS1

tan e n

e= ; — ~~ where k = tan"1— — — -

tan(e+k) cos S'

and tanS'-tanS cos 6

COMBINING THE REACTIONS In gear transmissionhaweA
FROM ONE PAIR TO ANOTHER, between two elements, #1

and «2,

f m w = angular velocity

hence ;r- = 8t — — Likewise between gear elements
Ti wt

#3 and #4, -p w

Then if gear #2 is on same shaft as gear |3, we

have T = T and w = w hence

23 ? 3

T T w w

42 's-1

- x — = 6 — E — —

T3 Tl *4^4

= e e —

ia 34

289

w

Now the velocity ratio — may be obtained as outlined

w

in previous discussipn on velocity ratios.

In the proceeding discussion the inertia effect
of the gear elements has been neglected in comparison
with the friction developed between the gears.

TORQUE AND POWER REQUIREMENTS In elevating ertravers-
FOR ELEVATING AND TRAVERSING ing a gun, we nave three
MECHANISMS. important periods :-(a)

accelerating period,
(b) the period of uniform

motion and (c) the retardation period. The maximum
torque obviously occurs during the acceleration and
power is continued through period (b), while the
friction of the mechanism brings the system to rest

during period (c).

Let 1^ = moment of inertia about the trunnions of

the tipping parts.
1^ = moment of inertia about the vertical

traversing pivot of the tipping parts

and top carriage.

E = elevating gear (tangential reaction.)
J - radius of elevating arc.
r = radius of traversing arc.
Mt~ friction moment of trunnions

M^= friction moment of traversing pivot.
Then during the acceleration,

J t t dt« for eievating the gun

E.r-M'=l' ill

fc * dt for traversing the gun

Now MI and M^ are constant depending approximately
on the weight on the bearing, while on the other hand
E and E' depends on the elevating or traversing
motor characteristics.

290

Neglecting the inertia of the gear elements, we
have, the torque transmitted varying directly as the
number of teeth, that is between any two gear elements,

T i

for gear pair 1-2

3
— = -- - for gear pair 3-4

Ten

4 34 «

for gear pair 7-8

If gears 2 and 3, 4 and 5, 6 and 7 are assuned on
sane respective shafts,

T, = T3, T4= Ts, T6= T7

then

Ii . L . L. , !i . L = A „ i. , L. „ ^i,J_ . ^_ ,

T. T, T. T. T. °2 "4 ". ", Si '.4

e t

5 • 7t

Now TV = E re and = * * *

e e e e e

t t 34 S 8 78

then gr n. n n n

T = — - ( — - x -i x -5. x — 1)
« na n4 ne "«

hence T „ n n n d20

e — — ( x — i x — i x — S)J-M 4. = !+ f°r elevating

re ni na n, nr dt

the gun.

291

l 2 4 s « II

e — ( — * — * — * — )r-M* = It — - for traversing

' * z

the gun.

dtz

and for the sngular velocity ratios, we have,

J
and w = — wt; - for spur or bevel gears: (elevating;

r i
* = — *t:~ ^or 3Pur or tevel gears: (traversing)

re

= -2— wt : for worm gear in contact with
e np

elevatin arc (elevating)

2nr

. = -TP— wt : for worn gear in contact with
• n p ••

traversing arc (traversing;

CHAPTER V.
RECOIL HYDRODYNAMICS.

OBJECT. The modern recoil system is

essentially a hydropneunatic device
for dissipating the energy of recoil
by so called hydraulic throttling
losses, and returning by means of the
potential energy stored up in. the compression of air,
the recoiling mass into battery. The potential
energy at the end of recoil required to return the
piece into battery is relatively small compared vritb
the energy dissipated by the hydraulic braking.
Further the potential energy of counter recoil is in
greater part dissipated by the hydraulic counter re-
coil buffer in the return of the recoiling mass into
battery.

In the design of the braking system misunder-
standing may result due to incomplete comprehension
of the fundamental principles underlying the hydraulic
throttling and the various hydraulic reactions. Hence,
in this chapter a resume of the essential principles
underlying the hydraulic phase of recoil design will
be attempted.

ELEMENTARY HYDRAULIC Consider an ordinary tension
BRAKE brake (fig.l) the oil being

throttled through apertures in
the brake cylinder from the
front or rod side of the piston
to its rear.

Let ax = area of the variable apertures or

orifice .

An = effective area of piston on rod side.
A = total area of cylinder.
ar = area of rod.
Pn * total hydraulic pull.

293

294

u

Dp

L

J

CvJ

00

L

295

Vx=velocity of recoil at displacement x.
vx= velocity of oil through apertures.
D = weight of fluid per unit volume^
p = p^= intensity of hydraulic pressure.
C = contraction coefficient of orifice.
K = reciprocal of contraction coefficient.
For a displacement dx, the mass of liquid moved
by the displacement of the piston, becomes,

D Ah dx

and due to the contraction of the liquid

g

in the throttling aperture or orifice, its

effective area is reduced to C ax, therefore, the
mass is accelerated to a velocity

A v \r A w

Hh vx Hh Yx 1

vx = = , since K = -5— now the energy

SLy £L«

of the jet,

* D Ah dx .

vx becomes, dissipated by a

loss due to sudden expansion

in fhe rear part of tlie cylinder, where we find a void
equal to: (A-Aj1)x= arx . By the principle of virtual

work, evidently x A, v

- D Ah dx *h vx

p* d«- -J^- (T^->

hence 1 D Au V*

g c a x

that is in terms of the liquid pressure
: D K« A> V

Consider again a brake where the throttling
occurs between the hydraulic cylinder A and a re-
cuperator cylinder B containing a floating piston
which is contact with the oil on one side and the
air on the other. See fig. (2).

Let p = pressure intensity against hydraulic
or recoil piston.

296

Aj, = effective area of hydraulic piston.

ax = throttling area between the two cylinders
which we may assume is controlled by a

spring.

vx = velocity through orifice.
Vx = velocity of recoil.
Va = velocity of floating piston.
Aa = area of floating piston.
pa = pressure intensity against floating pia-

ton.

x = displacement of floating piston.

Then by the law of continuity, A^ dx = Aa dx
Due to the contraction and sudden expansion of the
liquid from the throttling apertures, the loss due
to eddy currents becomes,

D A dx «h "x

By the principle of virtual work, we have,

r D Ah dx Ah Vx „

, ,1 2 i) / U A \l

Ph Ah dx -paAadx = - ( - )

g C ax

Neglecting the slight change in the total kinetic
energy of the liquid in its virtual displacement.
Simplifying, we obtain,

p K' v

g a x

which gives the drop in pressure through the orifice,
or the so called throttling drop, Obviously, Pn=Ph*h»

as before, i

- D K*AV

(4)

PRINCIPLES OF (1) Though in the analysis
HYDRODYNAMICS. of recoil brakes, liquid viscosity

is an item of importance, the

viscosity effect in modifying pressures is, with a
few exceptions, small, and therefore, for a first

297

approximation we will consider an ideal fluid, that is
a liquid with no viscosity.

(2) It may be shown by simple analysis in the
consideration of a small tetrahedron or triangular
prism that the pressure intensity on all planes at a
given point within a fluid is the same, the bodily
forces such as gravity, inertia resistance etc. in
limit being eliminated since they are functions of high-
er order (three dimensions) than the surface pressures
(two dimensions).

(3) By higher analysis it may be shown that
fluids flow in so called stream lines and therefore
the variation of pressure with velocity at various
points along the stream line as well as the change
in such due to the acceleration of the fluid as a
whole, may be determined by a consideration of the
pressures on continuous differential elements. Due

to the mutual action between differential elements, we
nay, by simple integration along a stream line determine
the pressures at the extremities of a stream line tube,
that is the end pressures as well as the terminal
velocities.

Consider a differential element A 8 C D along a
stream line, of cross section w of length ds and a
circumferential perimeter c.

Let, the intensity of pressure on A D be p, the
weight per unit volume be G, then for the pressures on
the surface A B - C D and the wall of the tube, we

have

dp

pw-(p+ — ,ds)(w-dw)-pcds sin <*-D " ds sin J0 =
ds

— but cds sin <x = dw. Simplifying and

g dt

dividing through by w, we have, - dp-D ds sin 0 -

2*1*1 (5)

g dt

dv s dv dv dv dv
but dv = — dt + — ds hence — = — + v -— -which
dt ds dt dt ds

shows the acceleration is both a time and space

298

function, inserting in (5) we obtain,

Dds dv dv

-dp-D ds sin 0 (77 * * 7~> <6)

g dt ds

Integrating from (1) to (2) along a stream line, since
the mutual reactions between contiguous particles can-
eel out, we have,

; i 'd°<zi>* 4?1

1 ,dv v"v

Obviously,

/ * ds sin 0 =Z -Z

hence

dv

The term / ds — is of special interest and when it
dt

occurs the motion is not steady. This tern is

theoretical, always existing in a recoil brake, since
the fluid in addition to a space variation of velocity
due to changes of sections, is on the whole ac-
celerated as well.

. , dv dv

To evaluate / ds — it is necessary to express -—

dt dt

as a function of s. If now we assume the same stream
lines to exist whether accelerated or under uniform
steady motion, we have, by the equation, of continuity,

wiVi = *2V2= *3 V3 and

dvt dv^ dva

1 dt dt ' dt
hence knowing the acceleration at one section,

dvn w! dv

—-— = — r— for any point "n", hence if w is a con-
dt wn u t

tinuous function of s, we have
dv 1 dvt, 1

wt - hence the line integral
dt w dt dt of the acceleration

along a stream lines, becomes,

299

dv d_v ds
3 dt " "ldt f(s)

The line integral of the acceleration may be
obtained to a sufficient degree of exactness by
dividing stream lines into a linear group of columns
of various sections, obtaining the proper acceleration.
To form (8) and multiplying by the length of the res-

pective columns and then adding these columns together.

1 dv

The term - / ds (— — ) is found usually to be relatively
g dt

small compared with the pressure drops due to throttling
and the changes of pressure due to changes of section.

Hence (7) reduces to the energy equation for
uniform or steady flow, known as Beraoulliis theorem,
that is,

t t

P T

D 2 D z 2g
p v*

The term — *• +Zt+ — x is known as the total head at

section (1), composed res-

pectively of a pressure head, gravitational head and
a velocity head,

(4) When friction, viscosity or turbulent
motion occurs Bernoulliis theorem is modified by a
friction head hf.

Considering a tube of a stream, we have for steady
motion Dw ds

Pt*td«^t!ftd«,+D"t<Ut(*t-Zt)-ar = * * (v*-v*) (10)

2 2g

Where d Wf corresponds to the differential work
due to friction for a differential quantity of flow
d ft.

By the equation of continuity, dQ = wtdsi= ",dsf

hence (9) reduces to

Pt-pt+D(zt-zt)-^»-(vX> (ID

dWf i

but - •>- = hf i , known as the head loss due to
dft 2

friction between 1 and 2, hence

300

t i

<r * 5 * z.>- (r * if * z.) • h<;

It is to be especially noted that in the flow of a
liquid through an orifice as in a recoil brake, the
•ajor loss is in the nature of a frictional loss of
head due to the contraction and sudden expansion of
the liquid through the orifice, thus

u

" * =

in equation (3), that is

p,-p. r "'

— - - = - 7 since Z = Z approximately, also
*a

t x

Vt V2

— and - are relatively small.
2g 2g

It may be shown by a somewhat similar analysis
that in the consideration of friction of or turbulent

loss of head due to throttling, that, from (7) we
have

2 8

where if wo = the area of the jet

VQ = the velocity of the jet.
where hfi has the form,

Hence anaxact expression for a stream line
passing through a jet, and the whole stream line
itself under acceleration, becomes,

D 2g D 2g g dt w0

(5) The pressure variation across a stream
line may be obtained by a consideration of a cylinder

301

the end faces of which are in the outer and inner
boundary surface of a stream line tube, and the axle

.is perpendicular to the stream line axis. We have,
if w= cross section of differential cylinder
r = radius of curvature of stream line

d = height of differential cylinder
0 = angle between r and the vertical

that a

Dw v
(p+dp)w-pw+Dwdr cos 0 = — y dr — (15)

2
V

wdp = Dwdr( -- cos 0)
gr

j g

hence -7*- = D (- -- cos0) (16)

dr gr

which given the rate of change of the pressure across
a stream line with respect to the radius of curvature.
Neglecting the weight component, we have,

£ - —

dt gr

Hence for circular or vortex motion, the change in
pressure along the radius, becomes,

dr

In particular if the total system acquires the

sane angular velocity1"'- , we have,
— dt

1_ = r (*i)«

r dt

since the total head at any point in a fluid equals,

2

p v

H = ~ + Z + r— the variation of head across
D 2g

dp vdv
a stream line becomes, dH. - —— +dZ+ -

L * vdv \ (2o)

= r*- +dr COS0+ - )

D g

Substituting Eq.(16) in (29) we "have,

302

_• vdv

dH= — dr + (21)

8* 6

which is the general equation for the change in head
across a stream line.

(6) When the flow is radial, evidently the
flow outward fro» circumferences of various radii,
becones, Q » 2n rv = 2* rv , hence voro»vr

or _

v ro

— = — and for steady motion, we have,

v_ r
o x 2

!°. + Is + - L + L. , z

D 2g 0= D + 2g

hence * *

°/, *i /««x

P-P0= 2ja~ 7») (22)

In terms of the total head H, we have,

* 2

P vo ro

r • H - -3^- (23)

(7) A free circular vortex occurs when the
total head of any annular stream line of the vortex
is the same.

That is, for any annular stream line,

t

p v
fl = -jj— + — — = const.

To find the distribution of pressure, we have,

dp vdv

~~ "* ° and for the flow slowly out-
u g

ward radially, we have,

P-Po vo"v*

— - — « •»- (Neglecting friction)
D 2g

Thus the pressure variation is exactly similar
to that of ordinary radial flow.
How from (21) since dH » 0

303

v vdv dv dr

— dr + = 0 hence — = and from the la*

gr g v r

of continuity

for the flow outward, v r » v r hence v - JL

r

Likewise the flow outward is exactly similar
to ordinary radial flow.

THE EFFECT OF THE The viscosity of a fluid is

VISCOSITY OF FLUIDS. the shear stress to the distortion

of the fluid and this stress is
measured by the coefficient of
viscosity times the rate of

distortion. In other words the viscosity or coefficient

of viscosity, becomes,

s dv

u = — or s = u jr where v * velocity of a

— lamina flow (ft/sec)

h = normal to flow

lamina (ft)
s 3 shear
1° Flow between flat surfaces:

r

dv

S = u — .bl
dh

d2v

dS*u rrr dh.bl
dh'

Now considering the- forces on a lamina of thick-
ness dh, breadth b and length 1, we have, for a constant
pressure head (pt~p )

(p ~P ) bdh-dS = 0 for uniform flow

304

(p -p )bdh-bln ^-7 dh * 0
dh

d'v

(pt-Pt>-lu —7

dv

— »

dv (Pt-P.

Integrating, us have — » ' + C

dv
when — = 0, Ct = 0

(p.-pX

Integrating again, v = — - — - - * C. when v * 0,

2ul

(Pl-P,)Hf
and Cg = - - Hence the distribution

of velocity across a
section is given by the equation,

<Pt-Pt> * ^

v = ^ul - (^ "" 7"^ (ft/sec) as measured from the

center. For a dif-
ferential flow, we have

P -P, 2 H*
dQ = vbdh = ~ — - (h -- ) bdh and for the

2ul total flow,

summing up oo both sides of center line, we have,

Q = — - bH Therefore the drop of pres-
12 ul

sure between flat surfaces

in a rectangular channel becomes.

12ul A ,., N
pt-p2 = - ft (Ibs)
bH»

For the particular case of a square section,

12ulQ

Pi-Pa = "TT" (Ibs)
n

305

2° Plow through a circular section:

p, 1

p

1 1

1

1

1 2-
I rh

1

<L
i

dv

The viscosity shear becomes, S = - 2nrlu —

dr
(r ID

dr

dS = - 2nuld — dr

dr

Considering the forces on a cylindrical lamina of
thickness dr and length. 1, we have,
(pt-p2 = 2iirdr - dS1 = 0 for uniform flow

AI d^
T— )

(pt-p,)r*ul dr r = 0

Integrating, we have ^p _p )r*

r — = - — - — + C

dr 2ul

which may be written

dr 2ul r

dv C

when — =0, r = 0, hence — * 0 i.e. C =0 since
dr r

r may have any finite value. Integrating again

v = - — — — — * C-2 when v = 0 for the boundary
4ul

surface,

r » R

306

t,
and v = -i-i (R*-rf)

which gives the variation of the velocity over a

cross section as a function of the radius from
the center.

For the total flow per second, we have

r

Q » / 2nrdr.v
o

/ (R'-r')rdr

2 ul o

(pt-Pt)nR4

8 ul

Hence for the drop of pressure through a small orifice
where there is no abrupt change in section,

8 ul
Pi~pa = — 4~" Q where u = coefficient of viscosity.

71 R

that is the drop of pressure varies as the length and
inversely as the 4th power of the diameter of the
orifice.

PRINCIPLE OF MOMENTUM The various formulae
AND DYNAMIC REACTIONS. previously developed de-
pended upon the ap-
plication of Bernoulli's
theorem, or the energy equation

of hydro dynamics. A theorem of equal importance is
the principle of momentum and from it with a com-
bination of Bernoulli's theorem, we may compute the
various dynamic reactions, that occur in hydro
dynamic problems.

If P = the unbalanced reaction on a mass of
water and the velocities of the mass is changed in

307

time t, from vt to V2 ft/sec, then Pt=ra(v -v ) . In
the application of this principle

(1) the mutual reactions between the
particles of water are obviously en-
tirely neglected.

(2) the velocity components in the
direction of motion are only to be
considered.

DERIVATION OF

THROTTLING

FORMULAS.

(1)

Loss of Head due to

sudden expansion.
In the flow of a fluid through
an orifice the drop in pres-
sure or loss of head, is

primarily due to the sudden expansion or abrupt
change from a small to a large section, of the fluid
flow. The loss of head is due to the formation of
eddies due to the sudden expansion of the flow and
the consequent dissipation of energy. If the cross
section of the flow is gradually enlarged from that
of a small orifice to a large section, no eddies are
produced and we have no loss of head. Thus in the
Venturi meter the fluid passes from a large section
to a very small section and then back again to a
large section but since the change in section is
gradual, we have no drop in pressure. Hence we have
a very important principle that is fundamental in
the design of recoil throttling orifices:-

A throttling drop in pressure cannot be pro-

duced without a sudden change in section of
the flow of the fluid.

Y*\//t///jfa//l

308

Consider a flow of fuild passing through

sections, ab, mn and cd respectively. Let the
cross sections of the stream be w at ab w at cd

1 12

and the corresponding pressures be pt and p res-
pectively. Let pQ be the pressure at mnt (Ibs/sq.ft).

From the energy equation, we have,

t *

P! Vl P2 Vt

— + — = — + — + hj assuming a continuous
D 2g D 2g uniform flow.

From the principle of momentum, we have

•

Q

Piwi*Po(w2~wi)~P2W2= g~ D (v2~vi) where Q = the rate

of flow (cu. ft/sec)

D = the density of fluid (Ibs/cu.ft). Now from the
experiment it is found that po = pt hence the momentum
equation reduces to,

ft _, Pi P2 T2(V2-Vl)

(P-P" D(v-v) and— — =

at
hence

g
(v-v)2

which simplifies to

tt

hf= - or in terms of the area of the orifice
2g

and the enlarged section, since wtvt = *2v2

«' 2'

T W V W

hf = -i- (1- -i)2 = -i (-1 - 1)
2* w2 2g wt

(2) Loss of Head due to sudden contraction,
When the cross section is suddenly diminished
beyond the reduced section we have eddying of the
flow with a resultant loss of head. This too is
really a special case of (1) since just beyond the
contracted section, the stream becomes even more
contracted, followed by a sudden expansion until
the stream reaches the cross section of the con-
tracted area.

309

Therefore if w = the area of the contracted section,
then the cross section of the contracted stream be-
comes, c w where c = depends upon the preceeding
area w and the area of the orifice. In terms of the
velocity of the orifice, the loss head equals

1 2

V W V

hf= — — ( - 1) = E-r — where if w = the cross

2g ex 2g

section before the sudden

contraction of section, the experinents of Weisbach
give for

o.i o. 624 o. 360

0.2 0.632 0.340

0.30 0.643 0.310

0.40 0.659 0.266

0.5O 0.681 O.220

O.6O O.V12 0.162

o.7o o.'/BS 0.106

O.8O 0.813 0.053

0.90 0.892 0.014

l.OO l.OO O.

IB the special case when — — = 0, that is the area of
the orifice w± is entirely negligible with the flow
from the large cylinder as in a flow from a resevoir,
c = 0.6 and E = 0.445.

It is to be noted that the above analysis holds
only when the length of orifice is sufficiently long
to allow the contracted stream in the orifice to ex-
pand and completely fill the orifice before expanding
in the region beyond the orifice. Hence the loss of
head due to sudden contraction only holds for long
orifices or entrances into long channel parts.

310

LOSS OP HEAD AND PRESSURE Assuming uniform flow
DROP THROUGH RECOIL ORIFICE. from the recoil cylinder

of effective area w,
through an orifice of
cross section wt dis-
charging into a cylinder or channel of cross section
w . Then from section w to the mid section of the
orifice,

v* P v2

5 + jj- - -5 + jj + hfc (1) hfc = loss of head

due to sudden
contraction

and from the mid section of the orifice w to the
rear of cylinder or channel cross section vr ,

2 a

^ * 5? = ^D + 55 + hfe (2) hf e = loss of head

due to sudden
expansion.
Adding (1) and (2), we have

P / P2 v*

— + — = — + — -«• hfc + "f e

D 2g D 2?

Very often v = va approximately and usually the heads
corresponding to v and v are small compared with
the pressure and throttling heads and therefore the
velocity heads may be entirely neglected. We have
then,

P - P.
1— = hfc + hfe that is the drop in

pressure through an orifice
is equal to the

total head lost due to sudden contraction and ex-
pansion.

Now

?' 2'

V W .V VI .

hf - -i(-± - D'= ~U- -1)2
2g w Xg w2

311

*f* * *T*

further AV = w^* wfvf, (where V- the
velocity of the recoil piston.)
(w=A = effective area of recoil piston)
hence

A*Vf< w

^r(1-^

When the orifice is in grooves in the cylinder or through
orifices in the piston, we have

)since *t=A approx-
iraately,

v =V approximately and

"^ * c7" aPProximate1^

(

)the effective area
(of the recoil pis-
)ton and wt=effective

j^

Therefore if n - — ,we have

V . ,.t'
— (m-1)

(area of orifice,

)that is the contracted

(flow through the

)orif ice.

(c = coefficient of

)contraction of

"*

(orifice.

Usually the loss of head due to sudden contraction
nay be entirely neglected as compared with the loss
of head due to sudden expansion, hence we have, for
a very close approximation of the pressure drop,
(1) With throttling through grooves
in cylinder or piston,

p-p V8
2 (m-i)»

D 2g

or

.*'«*'
A V

cw

both forms having useful applications.

of
contraction,

312

(2) With throttling through an orifice
froa the recoil cylinder to the re-
cuperator,

»i « area of orifice

w - area of channel leading from orifice.

ANALYSIS OF THE MUTUAL In an ordinary brake
REACTION IN A RECOIL BRAKE. cylinder we have a groove

in the cylinder or at the
circumference of the re-
coil piston.

As the recoil rod pulls out, a pressure is created on
the front side of the piston, due to the forcing of
the fluid through the orifice groove. The pres-
sure is by Berboull's theoren, obviously lowered in
the vicinity of the orifice due to the increased
velocity of the flow.

Hence, with an orifice in the piston, the
sure is not uniformly distributed over the

effective area of the piston.

Therefore, the brake reaction is not equal to the
product of the recoil cylinder and the effective area
of the recoil piston.

Let X = the total reaction of the fluid on the re-
coil piston. (Ibs)

p * the pressure in the recoil cylinder. (Ibs/sq.

ft)
A « effective area of recoil piston.

= 0.7854 (D» - d£ ) (sq.ft)
where Dr = dian. of recoil cylinder (ft)

dr » diam. of recoil rod (ft)

Ar * area of recoil brake cylinder = 0.786 Isq.ft)
V » velocity of the recoil piston (ft/sec)
v * velocity of flow through the orifice, (ft/sec)
D » density of fluid (Ibs/cu.ft)
w » area of orifice (sq.ft)

313

c = contraction factor of orifice.

Assume the recoil rod to only extend from one end
of the piston. In this case, we have a void in the
rear of the piston due to the volume displacement in
the front of the piston being less than in the rear
of the piston.

Assuming the pressure in the orifice to be small,
we have, for the reactions on the fluid from front
head of cylinder to a cross section at the center
of the orifice:-

(1)

and the reactions on the fluid from the orifice to
the rear head of cylinder, becomes,

The reaction on the piston = X to the rear

The reaction on the cylinder = pA - Y to the front

Adding (1) and (2) we have, pA> - X-Y = 0, which
is immediately obtained since there is no change in
the total momentum of the fluid, as we should expect
from first principles, since the fluid acts as a medium
for the transmission of the reaction between the re-
coil cylinder and the recoil piston. Hence pA - Y = X.

which gives the actual
reaction exerted on the

"brake piston. Since C vr v = A V, by the law of con-

tinuity, then

£ 2

AV Cwv , v D A v , .
v - — and V = - and X = pA - - (ibs)
cw

Dv*
now p = , from Bernoulli's theorem,

314

Hence X = — — (A - 2 cv»)
2g

= ° \ I (A-2 cw) (Ibs)
2gc w

D v* DA* Y2
but p = - = - hence X = p(A-2 cw)

2g 2gcawa

That, is the reaction on the piston equals the
product of the pressure in the recoil cylinder and
the effective area of the recoil piston, where the
effective area of the recoil piston equals the an-
nular area betneen the recoil cylinder and piston
rod decreased by twice the contracted area of the
orifice.

A physical explanation i$ that due to the
pressure of the orifice, we have the pressure lowered
around the orifice. Hence we must not only subtract
the area of the orifice, but also an additional
equivalent area which is to account for the lowered
pressure about the orifice.

Since c = 0.6 approx., then 2 cw = w approx.,
and therefore for practical calculations, the an-
nular area of the recoil piston is merely decreased
by the total throttling area through the piston.

Ifhen the rod is assumed to extend through both
ends of the recoil cylinder, we have a continuous
rod in the cylinder and therefore no void is pro-
duced during the recoil.

Assuming the same symbols as before, we have,
since the total change of momentum of the fluid
is nil, pA - X - Y = 0. Hence X = pA - Y and the
fluid merely transmits the mutual reactions be-
tween the recoil cylinder and recoil piston.

Let pw = the pressure in the orifice. (Ibs/sq.ft)

) Xf = total reaction
( on front of re-
) coil piston.
for the momentum of the fluid contained in the front

315

part of the cylinder to the orifice, and

DAY
Y-pww-Xr= - v ) Xf = total reaction on

( rear of recoil

) piston

for the momentum of the fluid contained from the
orifice to the rear end of the cylinder. Now
X = Xf-Xr= total reaction on recoil piston. Due
to the sudden expansion of the fluid after leaving
the orifice, the pressure on the rear face of the
piston, becomes, pw(A-w)=Xr (assumption from ex-
periment - sudden expansion), hence

Y -pwA = D A V v and X. = Xf-pw(A-w)

DAY
=(p-pw)A -- - — v

Dv*
Applying Bernoullis' theorem, we have p-pw = -

but by the law of
continuity c w v = A V therefore

Dv*

= — — (A-2-cw) (Ibs) Since pw is negligible
*f

compared with p, we have

Dv2
P - Pw - P =

2g

hence, as before X = p(A-2cw)

That is the total reaction on the recoil pis-
ton equals the product of the pressure in the re-
coil cylinder and the effective area of the re-
coil piston, when the effective area of the recoil
piston equals the annular area between the recoil
cylinder and piston rod decreased by twice the contracted

316

area of the orifice.

Since c = 0.6, 2cw = w approx., and therefore
again for practical calculations, the annular area
of the recoil piston is merely decreased by the total
throttling area through the recoil piston.

DERIVATION OF RECOIL We may consider the throttling

THROTTLING FORMULAS, effected in either of the follow-
ing manners: (1) throttling
through grooves in the cylinder
wall or through a variable

orifice in the piston itself, -(2) throttling through

a stationary orifice.

(1) Throttling through a variable

orifice in the piston or grooves in

the cylinder walls.

Let

A = effective area of the piston, i. e. the
cross section of the cylinder minus the
cross section of the rod. (sq.ft)
p = the intensity of pressure at the pressure

end of the cylinder (Ibs/sq.ft)
D = the density of the liquid (Ibs/cu.ft)
V = the velocity of the recoil (ft/sec)
w = the area of the orifice (sq.ft)
v - the velocity of flow through the orifice.

(ft/sec)
X = the total fluid reaction against the

piston (Ibs)

Then, we have, (neglecting the small pressure in the
orifice) D A v

pA -X » v - - - for the momentum

generated in the jet,
Dvz
and p = _____ for the energy of the flow

in the jet.
AV = cwv ----- from the law of continuity of the

flow,
then

317

DA 2cw 2
X = ~ (1- — ),

3 2

DA V , 2cw,

(1 r-) (Ibs)

2gc*w2 A

Since the reaction on the cylinder is the

difference between the force pA at the pressure end
and the reaction of the jet

D A V

- v flowing from the orifice we have the reaction

on the cylinder also equal to
D A V DA3V* 2cw

as would be expected from the equality of action and
reaction.

Ulith a continuous piston rod through both ends of
the cylinder we may neglect the pressure through the
orifi*ce and since by experiment the pressure on the
rear face of the piston is practically that through
the orifice, the reaction on the piston remains the
same. Here again the reaction on the cylinder is

DAY .. D A V

pA-p A = pA -- v, since p A - - v as would

5 3

be expected from tlie equality of action and reaction.
The reaction X on the cylinder may be written

Y PA^2 2cw
X =

2gc*wa A '

Dv2

Further since p = = — • — , , we have also.

Zg 2gcaw*

X = p(A - 2 cw)

= p(A - w) approximately.

Thus, knowing the pressure in the pressure end
of the recoil cylinder to obtain the reaction on the
piston, we must multiply this pressure by the ef-
fective area of the piston minus the area of the re-
coil orifice.

(2) Throttling through a stationary

orifice.
With a stationary orifice, the throttling

318

usually takes place between the recoil or brake
and recuperator cylinders. The loss of head or pres-
sure drop is mainly due to the sudden expansion of the
flow from the orifice, though with a relatively long
orifice the loss due to sudden contraction may become
appreciable.
If

w = the area of the orifice (sq.ft)
A = the effective area of the recoil piston

(sg.ft)

V = the velocity of recoil (ft/sec)
v = the velocity through the orifice (ft/sec)
c = contraction factor of the orifice.
H = the area of the channel leading away from the

orifice, (sq.ft)
Then from Bernoulli's theorem, we have

p-pa ) where p = the pressure in the

~~~ = ^t ( recoil cylinder.

) pa = the pressure in the
( recuperator.

Mow ) hf = total head lost due to

hf=hfc+h.fe ( throttling.

) ^fc= l°ss °f head due to
( contraction.

) nfe~ l°ss °f head due to ex-
( pans ion.

T*

Now hf., = £ — where 5 may be taken 0.35 to 0.5 and
gf

and * z

v , cw.a v ... cw.* f.

hfe= "^ (1 ~ "1L) hence hf= ~~[(1 > * *]

2« 2g W

In recoil mechanisms W is usually made from 2.3 to 3.0
tines w. Then, we have, if c is taken approximately =
0.65

(1 --J.)* = 0.515 to 0.614
For flow from an orifice into a large reservoir

319

4 > 0 and (1 - £-)* < 1

n *

Hence usually

cw «
[(1 - -T- ) + &] = 1 approximately,

D A*V»

-55 hence p-p. = - for the drop of
2gcawz

pressure through

the orifice. The reaction on the recoil piston is,

D

X = pA = - — . . + p.A
**

In recoil design, it is customary to measure
areas in sq. inches and pressures in Ibs/sq.in.
Further the average specific gravity of the re-
coil oils used in our service may be taken at 0.849
and therefore the density D becomes, D = 62.5 x
0.849 (Ibs/cu.ft).

The recoil throttling formulas become, therefore
(1) For throttling through a variable

orifice in the piston or grooves in

the cylinder vralls:-

X =
P =

6 K2A»V2

(Ibs) w =
(Ibs/sq.in);

KA*V /6~

(sq.in)
(sq.in)

175 w2

13.2 /x
KAV

175 w*

where K = — = 1.6 to 1.3 approx.

6=1 •—: c =- 0.6 to 0.8 approx.

(2) For throttling through stationary

X =

orif ices:-

KAV

175

320

~P* * 175«* (Ibs/sq.in)

CW ft

where K = 1.6 to 1.3 approx. 6 =(1 -- ) + E

VARIATION OF THE THROTTLING ffe have seen the

CONSTANT IN THE RECOIL total braking on the

recoil piston may be
expressed, when
throttling through a

variable orifice in the piston or through grooves

in the cylinder, as

K« A» V«

and when throttling through a stationary orifice,
as

XIL I IV A V f •* * \
= a - + p.A (Ibs)
175w«

. 2cw .i cw.a _

where 6=1 -- - and o = (1 -- ) + £

Since w varies throughout the recoil, 6 and 6' must
also necessarily vary in the recoil. Calculations
with the omission of the term 6 or 6* have been
found slightly in error and this error has been
ascribed to variations in the contraction factor
of the orifice. The contraction factor may also
vary but it seems more probable that the error is due
to the omission of the term 6 or 61 .

With stationary orifices -^ and 5 can very-

H

often be neglected and therefore the variation in
the throttling constant can be neglected. With
throttling through the piston or by grooves in
the cylinders -2filL is small but not negligible^

hence with this type of throttling variations in
the orifice are more marked.

321

For a preliminary design 6 and 6' may be
assumed equal to unity; but on recoil analysis and
careful tests 6 and its variation in the recoil
should be taken into consideration.

CHAPTER VI
DYNAMICS OP RECOIL.

ELEMENTARY PRINCIPLES. The object of the recoil

is to reduce greatly the
stresses induced in the car-
riage. Without recoil, the
reactions brought on the

various parts of the carriage are direct functions
of the maximum powder force, which would require a
very massive carriage for guns of large caliber.
The mutual reactions created by the powder
gases between the gun and the projectile is of
very short duration compared with the time of recoil
and for a rough approximation nay be treated as an
impulsive reaction. Neglecting the mass of the pow-
der gases, we have /Pdt = mv and /Pdt = MV. Therefore
mv = MV, where m = mass of the projectile

M = mass of the recoiling parts
v = velocity of projectile
V = velocity of recoil
/Pdt = impulsive reaction of the powder

gases.

The momentum generated by the action of the pow-
der gases in the projectile and gun is the same, as
is immediately obvious from the principle of con-
servation of momentum. It is to "be further noted
that finite forces, as the resistance to recoil, can
be neglected in the consideration of impulsive actions,
and since the generated velocity of recoil acts for a
differential time, the recoil displacement during the
impulsive action can also "be neglected.

The kinetic energy of the recoiling parts, after
the impulsive action, is

A. ~± MV'
Since V = , the recoil energy in terms of the

323

324

IB

velocity of the projectile becomes, A = — (- mv ).
Hence the energy of recoil is but

n

- of the energy of the projectile.

The total energy generated by the impulsive
action of the powder gases, is, therefore

i m

- (i . 5>«

Obviously the greater M, the smaller the energy
of recoil.

The reaction R between the gun and raount for a
recoil displacement b, is - MVa

R = ^

or in teras of the velocity of the projectile
_

"* , * * \

~H (; BV }

The reaction is thereby reduced proportionally
to the increase of ths recoiling mass M. Hence to
reduce the recoil reaction we increase the recoiling
mass 14 and the length of recoil h .

The dynamical relations for an elementary recoil
analysis in terns of the relative velocity of the
projectile with respect to the gun vp can "be readilj
obtained as follows:-

Vp = v + V assuming V measured in the direction
of recoil from the conservation of
momentum m Vg

MV * mv = m(vR- V): hence V = -

M + m

The energy of recoil is

and the recoil reaction

If the recoiling parts are hrought to rest hy
friction alone, R = u Mg

325

1 V2
hence b = - — • 3

2 ug

DOUBLE RECOIL SYSTEM:

When a gun is mounted on a movable mount as a car
body or itself rolls along a plane, we have virtually
a doubl.e recoil systen, the upper recoil being between
the gun and mount, and the lower between the mount and
plane. As a first approximation we will neglect the
resistance between the mount and plane as small com-
pared with the upper recoil resistance. Let

MR = mass of upper recoiling parts

MC = mass of lower recoiling parts

ra = mass of the projectile

vo = the muzzle velocity of the projectile

V = the initial velocity of the recoiling parts

v = the velocity of combined recoil

Then, during the impulsive action, neglecting the mass
of the projectile, we have,

T

for the projectile / Pdt = mvo (1)

T T

for upper recoiling parts / Pdt - / Rdt = MV (2)

o o

Where F is the vertical reaction between the upper
and lower recoiling parts. T

How R is a finite force, .*. / Rdt - 0, if t is

o

very small. Further the displacement of the upper
and lower recoiling parts inappreciable, since

T T

/ Vdt = 0 and / Fdt = 0 respectively
o o

Hence, nvo = MpV with no appreciable displace-
ment of either the tipper or lower recoiling parts and
no moraetitura imparted to the lower recoiling parts.
During the recoil, after the impulsive action, we have

326

T

for the upper recoiling parts / Rdt=MR(V-v)

o

T

for the lower recoiling parts / Rdt=Mcv

o

hence, the combined velocity of the system when the
relative recoil between the upper and lower recoiling

parts ceases, is

MRV

v = T. : —

If the mutual recoil reaction R between upper and
lower recoiling parts is made constant, then

v" c n V

R = Mc -— or T = — - where T is the

time of the
relative recoil. The relative displacement Z is,

,V+v v" V
? 99

£t & 6

Substituting for T, we have

McMR V*

Z 3 — — — • for the relative displacement
MR + Mc 2R

The relative displacement can also be obtained
from a consideration of the energy relations in
the recoil. We have

V 1 X _*

T) = — **R(V -v ) for the upper recoil-
parts

V i a

-T~ T) = - Mcv for the lower recoil—

parts

Subtracting:
RZ = J MR(Va-v2)- f Mcv*

that is the energy of recoil, j MRV =

is dissipated in friction and throttling (RZ) and

327

remainder is the kinetic energy of the combined
masses. Now since, M v
- R

= MR+*c
we have » *

i MRMc t
• « MR+MC

Therefore as before, the relative displacement becomes

MR+MC 2R

ELEMENTARY RELATIONS. During the travel of the

projectile in the bore of the
gun, neglecting for a rough
approximation the mass of the
powder gases, a mutual reaction

is created "by the powder gases between the gun and
projectile, which generates equal momentum in both
projectile and gun provided no extraneous forces
are exerted on the gun. The resistance of the recoil
brake is very small compared with the powder force,
therefore its momentum effect is negligible. After
the projectile leaves the bore, further expansion of
the gases take place and the reaction due to the
momentum generated in these gases causes an addition*
al increment in momentum of the gun. This additional
momenta is commonly known as the after effect of the
powder gases.

Assuming free recoil of the gun, if
m = mass of projectile
M = mass of the gun or recoiling parts
P = total powder reaction
v = absolute velocity of projectile
V = absolute velocity of gun in the recoil
u = relative velocity of projectile in bore
then during the travel up the bore / Pdt - mv = MV
but u = v + V for the relative velocity of the pro-

328

jectilc, hence m(u-7)=MV and the velocity of recoil

becomes

. m . mv

V = ( )u= —

m + M N

Since m is snail compared with M, we are not great-
ly in error in assuming u = v in approximate cal-
culations.

At the end of the travel of the projectile up
the bore, we have mv

and 7(

After the projectile leaves the bore if P = the
reaction exerted by the gases, then

** / it

/ Pdt = M(7f - V0) = nv where v = the mean

tg velocity of the gases

"m" after expansion. For a first approximation
v will "be assumed a function of the muzzle
velocity v0 and we will place BV =cvQm

Hence MVf=(m+cin)vo. For computations c will
be taken equal to 2.3. The energy of free recoil
becomes

hence

T. = i —

M

How the recoil brake exerts a resistance R through
a recoil displacement b, "hence

Rb= -MV* roughly,

and

R =

2M.b

The recoil reaction R is a measure of the stressed
condition of the carriage and very often for a given
carriage m, u, vo and b may one or all be changed.
To compare the recoil reactions, we have for the

sane gun, t t

Rt (•t*cit) v0i b,

,,_

and for R =R =R, then — = — where c =

\ (-.*«,)• v*,

2.3 approx., and for bt = bf, = "b, then

Rt (in^+cl^)3 v0±

r~ a ~, ., — : — where c = 2.3 approx.

R, (»,+co>t)* v«,

These equations are important in order to estimate
with a given change in the ballistics of a gun, the
necessary change in either the recoil or recoil
brake reaction.

The energy of recoil nay "be expressed as

m +cu, t _. a .

E = 1 r (m+cm) v0 }

M

r jM-f mvo) very roughly

m

= - (muzzle energy of the projectile) (approx.)

M

Therefore, to decrease the recoil energy M should "be
made as large as possible. Since further

The recoil reaction varies inversely as the recoil-
ing mass, and therefore to decrease R, M s"hould "be
made large.

EFFECT OP POWDER GASES The effect of the pow-
ON THE RECOIL. der gases on the recoil may

be considered during two
periods:- (1) while the pro-
jectile travels up the bore,

(2) after the projectile leaves the "bore and the ex-
pansion of the gases ta"kes place. In either case an
approximate assumption is- necessary in order to
represent the phenomena with sufficient simplicity.

During the travel of the shot up t"he bore it will
be assumed that the gases expand in parallel lamina,
and the motion of any differential lamina to be a
linear function of the distance. from the "base of

330

the "bore to the lamina in question, that is

i v + V

v = c s + c where c =» - V and c =

u

v = velocity of projectile
V = velocity of recoil
u 3 travel of projectile up the "bore
hence with free recoil

TO I

mv + 2 - v = MnV during the travel up the bore
u

but „

» i m ,u i . m(v-V)

2-v =-/ vds= —

u u 0 2

The equation of momentum of the system during the
travel up the bore becomes, therefore,

- (Y~V) (m*0.5S)v

• v + m — - — = MV or V =

MR+0.5m

Further since the relative velocity of the projectile
is

~ = v + V then, l»+0.5D(?r- V) =(M+0.55)V
at at

du
therefore (m+0.5m )T~

and for the displacement of recoil in terms of the
relative displacement of the projectile,

(m+0.5l)

M+V+I

If

P = the reaction of the powder gases on the

"base of the projectile
Pfc3 the reaction of the powder gases on the

base of the "bore of the g"un
then, for the powder gases, we have

I d(v-V) 5 d_v _ ra d_V (

^>" " 2 " dt Z dt ~ 2 dt

for the motion of the recoiling parts in free recoil,

331

Pb - *R ST ia4**' °ai^*a'1 ^"
and for the motion of the projectile

If tbe gun moves backwards a displacement X, while
the projectile moves forward an absolute displacement
x, then

X * / Vdt, x = / vdt (4)
Prom (2) and (3) in (1),

dV dv . i. dv _ £ dV
MR dt " dt 2 dt ~ 2 dt
hence

(Mp+0.51)— = (B+0.5 1)^ (5)

Integrating, we have as before,
(MR+0.5ii)V=(m+0.5i)v (6)

and.

(MR+O.SijX'dn+O.SSJx (7)

For the relative displacement

u = /l (V+v)dt or du = (v+V)dt
o

du (NR+0.65)d-u
V+v

. v . .... r /
. . X - / Vdt = / (

_-) du
o o Mp+tn+n

hence

m+0 . 5n

X * .. as was obtained by direct sub-

Mo +ra+m

stitution of displacements.

With a constant powder pressure during the travel
•up the bore, the time of travel becomes,

2u0 2u0(MR+0.5in) (2WR+m) UQ

* = — — = i = -••^^-•^

v+7 (t

Actually since the powder reaction varies during the

travel up the bore,

/U° °*u . fU° d"
o K 0

Since m and if are always small compared with Mp, we

have

,uo du
t = / 7— very closely

o

The relation between P^ and .P may be obtained as
follows:

m A ( v— V )

at * p

1 . dv

« ' T~ approximately

2 dt

hence

or

0.5—

Since however the linear motion of the powder gases
is an assTaraption', we "have more accurately,

dv
P-JJ = (ID + Bi) -— where for a first approximation

B = 0.5
The rngaa powder pressure lies "between P^ and P hence

Pffl = (1 + B -SL-) P where for a first approximation
8" = 0.3

ELEMENTARY ENERGY The Kinetic energy of the pow-
RELATIONS. der gases may also be considered a
summation of the elementary
energies of the differential
lamina. Assuming the gases to

move up the bore in parallel lamina, with the velocity
of any lamina a linear function of the end velocities
and neglecting the velocity of the gun as relatively

333

small compared with that of the projectile, we have,
for the kinetic energy of the powder gases,

i = total mass of powder

gas
u = travel up "bore of

projectile

where

• i s
hut v = -

•0 yS.O

v = velocity of any

given lamina
s = distance from "base

of Tbore to lamina

in question

1 /m * *

V » (3 )V

. 0+ : '

The Kinetic energy imparted to the recoiling parts

IS 22

1 (m+Q.Sm) v

ED= — —""-"•• "•••

2 M

"•.'• qi" ,• ' **

tooien*Qxe i>Ai \£ trevij ex »ic
Further if,

W = the potential energy of the gases at any

instant
P^ = the total reaction exerted on the treech

of the gun
P = the total reaction exerted on the base of

the projectile
X = the displacement of the gun in the direction

of its movement
x = the displacement of the gun in the direction

of its movement
Q = heat lost in radiation

J = the mechanical equivalent of heat
then, the equation of energy of the powder gases he-
comes

- PbdX - Pdx = d(Bp+W)+ JdQ

that is the external worfc on the powder gas system

goes into kinetic, potential or configuration

energy and lost heat energy. The above equation may

"be written -dW = P^dX + Pdx + dEp + JdQ

Further since PbdX = d(J (m*°'5i)' V> )

Pdx = d( mv* )

We have,

- „ . 4 [ i (("*°-Sii>\ .*|>,']« JdQ

* M 3

The work done on the system may "be represented by an
equivalent force Pm acting through a distance cor-
responding to the travel of the projectile up the
"bore, then - dW = Pm du + JdQ and since du = dx, very
closely, we have t

r((n+O.Sm) m , dv

Pm = t - * m + r 1 v —

M 3 du

Thus the equivalent mass of the system gun, projectile
and powder gases, referred to the displacement up the
bore is given "by the expansion,

(w+0.5i) i

M« = - + m + -

R 3

RECOIL AND BALLISTIC The recoil reaction, say, when
MEASUREMENTS. the gun is mounted on a ballistic

pendulum and the reaction of Vhe
projectile when fired into a
ballistic pendulum, differ by
fhe reaction caused by the ex-

pansion and consequent acceleration of the powder.
Obviously the snaller the charge the wore closely
would the swings of these pendulums "be alike.

BALLISTIC PENDULUM - QUM HOUHT8D OB PEHDULUM.

(a) When the powder charge is very
small, we have an equal impulsive
action on the projectile and gun.
If

d = the distance from the axis of rotation to

335

the center line of the bore.

M » the mass of the pendulum and gun combined,
k = radius of gyration about the axis of sus-
pension.

9 - angle turned by the pendulum
h » distance from the center of gravity to the

axis of suspension.

Then in consequence of the mutual impulse during t~he
fire, mv.d = Jfk*w and the initial angular velocity

is. therefore.

mv.d
w = — — — (rad/sec)

Hk» _,

d e «h

The subsequent motion is given by, — — = - *j- sin 8

Integrating,

.de.i 2gh

W =^COS e + c

de

when 6 = 0, cos 9=1 and— — = w

dt

therefore

t' 2gh

c = " -v~

and

,d8 » 2gh 2

(-rH = — ~ (cos 6-1) + w

U U 1C

Q w

At the maximum swing (— — •) * 0, and 6 = 9Q, hence

Q t

,

- cos e)

0

This is immediately evident from the equation

of energy, since ,, ,•

Hk w

= Mgh(l - cos e )

2 o

e0

The cliord of an arc radius "c" is 1 = 2c sin-—

2

e

Further since, 1 - cos 8Q = 2 sin -— "-
So mv.d

336

M It „ 9o

hence v = -- 2 sin —

n d 2

M k 1

= — — — v^gli
mac

whic"h means the velocity of the projectile approximate-
ly. The radius of gyration may readily "be obtained
experimentally by noting the time of swing.

(b ) When the powder charge is com-

para"ble with the weight of the pro-
jectile, we have to consider the

additional momentum generated by

the powder gases.

Assuming the center of gravity of the powder mass to
have a mean -velocity equal to one-half the velocity
of the projectile, we have

(1) during the travel up the bore,

•
(m+— •) v as the momentum

m

generated in the gun.

(2) after the projectile leaves the lore
we have an additional impulse p due
to the expansion of the gases.

Hence the equation for the motion of the "ballistic

pendulum becomes, - 2

d[(m-»- — )v+p] = Mk w

2/pT eo 1 IvTO

but w * — •— sin — » -- * —
k 2 c i(

1 , Mkl
hence (•+ — )v+p» — • /gh

2 cd

If now we repeat the experiment with the powder gases
as done in the experiments on the Ballistic Pendulum
"by Dr. Hutton, we "have

7V° * p *

where obviously V is greater than v,

337

S Mk(l-I0)

Subtracting, we have mv+-~-(v-v )= *• /gh

2 ° cd

i M (l-l«)fc
or v- -- (v_-v)= —•

2m in cd

To account for the powder gases experimentally, Dr.
Button proposed measuring with and without the pro-
jectile as follows:

Mkl

rov+p1 = — T /gh with the projectile

cd

hence (i-i)1c

o

— - -
m cd

The previous expression indicates this expression in
error "by the amount _

which for small charges is relatively small tut for
large charges may be appreciable and therefore can-
not be neglected. As an approximation, however, in
ordinary tests, the method of Dr. Button is suf-
ficiently accurate, for the measurement of the
velocity of the projectile.

BALLISTIC PEMDOLUM - IMPDLSE OP PROJECTILE

The "ballistic pendulum serves as a valuable
mechanical means of measuring the velocity of the
projectile though this method has been discarded in
modern practice. The dynamics involved is worthy
however of consideration in the general recoil pro-
"blem.

The time of penetration is sufficiently s~hort
for no appreciable movement of the pendulum.

Let d = the perpendicular distance from the axis
to the line of penetration of the pro—

338

jectile.
J » the distance from the axis to the position of

the projectile when the penetration ceases.
B » the angle between "d" and "J"

Then, the impulsive moment of the projectile Mp equals
the change in its angular momentum, hence
Mp * mv.d - mJ*w and the corresponding reaction on the
•pendulum "becomes M_ = Mkaw. Therefore mvd = (m"k*+-mJa )w
or mvJ cos B = (Mk*+mJ*)w. The initial energy of the
system consisting of the pendulum and projectile is,
therefore

«

w

and the worlc done by the weights in the movement to
the maximu swing, "becomes, Mgh(l-cos 6)+mgJ[cos B-
cos(9 - B)] hence, from the principle of energy, we
have,

j(Mk«+mJ«)w«=Mgh(l-cos e)+mgj[cos B-cos(6 -B)]
If B * 0, the equations reduce to mvJ =(Mk*+mJ*)w

(Mk«+mJ*)w*=2(Mgh-mgJ)(l-cos 8Q)
Combining these equations and noting that

eo

1 - cos 9=2 sin* —— , we have, for the initial
2

velocity of impact for the projectile,
2

+ mJ»)(Mh + mJ)g ] sin -
•J 2

GENERAL THEORY OF In the preceeding paragraphs the
RECOIL. theory of recoil was greatly

simplified by assuming the powder
period to "be of such short duration
as to be in the nature of an im-

pulsive action, and therefore the momentum of recoil
being generated practically instantaneously. In tha
theory of impulsive forces, we may neglect finite forces
such as the resistance to recoil since the time
of action is negligible. Further the displacement
in an impulsive action is entirely negligible. This
method gives fairly accurate results for long recoil

339

but when fbe recoil is shortened the results "by this
method of computation are only very approximate.

Fortunately due to considerable progress made
in interior "ballistics of late, the powder reaction
can be determined as a function of time and displace-
ment up the bore. It, therefore, "becomes a finite
force and the recoil problem during the powder period
can be treated with a considerable degree of accuracy.
Let Pjj = the total powder reaction on the breech
in Ibs. Its line of action is
necessarily along t"he axis of the bore.
B = the total braking due to the hydraulic

and recuperator pulls.
R - the total friction, (guide and packing

frictions) in Ibs.
K - the total resistance to recoil.
Hr= the mass of the recoiling parts
¥r= the weight of the recoiling mass in Ibs.
X = the displacement of the recoiling mass
from battery in the direction of the
glides.

0 = the angle of elevation of the g"un
a - the angle of the guides constraining the
recoiling mass with respect to the
horizontal .

From the theory of energy, we have the fundamental
principle:

The work done on the system consisting of the
recoiling part.g "hy t.hft pnariar gagfts must ptQiial tha
work dons on fha system T">y t^ift t. otal ra gi gt. anr*.?*. tin
recni 1 for t.hfi *»nf.i rr» rftnni 1 an nf./° t."ha enftrrfy r>f t.Vis
^ysteyn ar t,Tia bstfinning and °nd rvf rf.r'.DJ] i g 7.»rr>-

Froro this theorem we may prove that with a re-
sistance to recoil action throughout the powder
period, the energy which the powder imparts to the
recoiling mass whan free is always greater than the
energy which must be developed by the brafce in the
recoil. The greater the resistance to recoil during
the powder period the greater this deviation.

340

In the following proof the time effect of the
powder gases during free and constrained recoil is
assumed the sane or, in other words, the powder
reaction is regarded the same for any given time
whether the recoiling mass is contrained or free.
Theoretically of course due to the slightly
different motions in the two cases, the notion of the
powder gases themselves will be slightly different
and therefore a slightly different reaction on the
breech clock in the two cases. Since, however, the
difference in motions is so small and the powder re-
action so great, we may entirely neglect this fact
and assume the powder force to be entirely a function
of time and quite independent of the slightly different
motion in constrained and free recoil.

Supposing the gun to recoil along the axis of the
bore as is usually the case, the total resistance to
recoil evidently may be expressed as: K = B+R-Wr sin 0.

Therefore, the equation of motion for the re-
coiling mass for constrained recoil, becomes,

dV dvf

Pv - K = ID -T— and for free recoil, we have Ph=fflr -

dt

Integrating for any given time, evidently, V < V^

The work done by the powder for contrained recoil is
therefore less than with free recoil, since

*i t

Pt,V dt < / l Fb Vf dt where tt = the total time

00 of the powder

period. Kow the work done by the brake must equal the
work done by the powder gases in constrained recoil,

hence,

b t

/ Kdx = / PV dt

b t,

/ Kdx / PbVf dt

341

t b

but / * PbVfdt = j rarVf therefore / Kdx < j mr V*{
o o

that is, the braking energy or rather the work done by
the resistance to recoil provided the braking is effect-
ive during the powder period, is always less than the
free energy of recoil. When, however, no braking
resistance acts during the powder period, the work
done by the resistance to recoil or braking energy
must equal the free energy of recoil. Therefore, for
a given length of recoil, the recoil reaction is re-
duced by maintaining a resistance during the powder
period in a twofold way:

(1) due to the fact that gun recoils
over a greater distance, (i. e. the
displacement during the retardation

and in addition, the displacement during
the powder period),

(2) due to the fact that the braking
energy is always less than the free
energy of recoil.

In the design of a recoil system it is there-
fora, highly desirable to maintain a large resistance
to recoil during the powder period and thus effective-
ly to reduce the required braking and the consequent
stresses set up in the carriage, as well as to give
better stability to mobile mounts.

GENERAL EQUATIONS (1) When the direction of
OF RECOIL. recoil is not along the axis of

the bore. Consider the re-
coiling paris to be constrained
along guides or an inclined

plane making an angle "a" with the horizontal, and the
axis of the bore to make an angle 0 with the horizontal.

Neglecting the reaction of the projectile normal
to the bore, as small compared with the other reactions,
we have for the equation of motion for the recoiling
mass .

342

Pv, cos (0+a) - B - R - Wpsin a * m, -2.JL (D

dt«
hence

Pt, cos ( 0+a)-B-R-Wrsin a) dt » mp dv
and

/Pb cos (0+a)dt - /(B+R+Wrsin a)dt = mrv
but the powder force is measured by the rate of change
of momentum imparted to the recoiling mass when free,
that is dVf

P" ' 'r IT
hence Pb cos (0+a)dt » mr cos (0+a)d Vf

Substituting in the above equation, we have
mrVf cos (0+a) = /(B+R+Wr sin a)dt = mrV (2)
When the resistance to recoil is constant,
K = B + R + lfr sin a = a constant, and we have

Vf cos (0 + a) t = V (3)

Integrating again, we have,

Kt*

/Vf cos (0 + a) dt - — — = X

2mr

which gives the displacement from battery of the
recoil during the powder period, but
/Vf cos (0 + a) dt = E cos (0 + a) which is the
component displacement for free recoil in the
direction of recoil.

The constrained recoil at the end of the powder

period, becomea ?

KT

X = E' = E cos (0+a) - T — (4)

2mr

and the corresponding velocity at the end of the
powder period, becomes,

vr * Vf max.c°s^+a> ~— <5

where T is the time of the powder period.

Proa the energy equation in the motion from the
end of the powder period to the end of recoil, we

343

have ~ mpvp = K(b-xt) hence

j mr[Vfcos(0+a) -— ] = Kb-K[E cos(0+a)- j^-] 6)
Expanding and simplifying, we have

K[b-E cos('0+a)+VfT cos(0+a)= jmrVf cos*(£j+a)]

hence

t ,,a x , ., .

-mrVf cos (0+a)

K = b-(E-VfT)cos(0+a) (7)

or in terms of .the component reactions,

1 2' 2*

-,mrVf cos (0+a)

B+R+W. sin a = - (?')

b-(E-VfT)cos(«f+a)

where WVQ+ 4700 ^

Vf = - from the principle of linear
wr momentum.

E = total free movement of gun during powdei
period.

T = total time of powder period,

To deduce E and T we proceed as follows: (See Chapter
II) Calculate

rf Z1 Pm ti fa 27 ^N* -. i

b 3 uo[( TS r~ ~ 1} i /(1 " T^ r~) ~ 1 ]

16 Pe 16 Pe

where p^ = max. powder pressure X area of bore

and also,

~ b » Pbm :

then compute —

wv0+ 4700 w 2

Vf = - - - j V=

L *a J ...

344

where w - weight of projectile

w » weight of charge

wr = weight of recoiling parts

VQ = nuzzle velocity

The time of the travel of the projectile up the
"bore and the time during the expansion of the powder
gases are respectively:

b . 2u u 2(Vfl-Vfo) wr

*o - ; <*'3 log - * - * 8) tlo . — ^— _

3 uo
* - -- approx.

vo
Therefore the powder period, "becomes T = to+tlo

The free recoil displacement during the travel up
the bore, and during the expansion of the gases are
respectively:

u0(*+0.5w)

Therefore, the total free movement of gun during the
powder period, becomes, E = X fo +

MOTE: In the above and further formulae the units
employed are :

displacement in feet
velocity in feet per second
force in pounds
mass in pound units

With a void in the recoil cylinder during part
of the powder period, equation (7) becomes slightly
modified.

Let S = length of void in recoil cylinder

tg = time of free recoil to end of void
Neglecting,

R+Wr sin a as small compared with B, ws
find K =0, until distance S is reached in the recoil.

345

Therefore we "have

= E cos (0 + a) -

K(T-ts)
2*.

cos(0+d) -

K(T-ts)

(8)
(9)

where T - time of total powder period. Substituting
(8) and (9) in the energy equation,

-, mrvr = K(b-xt)
and simplifying, we have

1 2f 2'

- mrVfCOS (0+a)
K = b-CE-Vf(T-ts)]cos(0+d)
To evaluate ts, t"he time of recoil with void, we have

t. = -(2.3 log ^- +~ + 2)
a D D

.

where

(w+-£-)cos(0+a)

- D± /l- ~)
16 = P

(11)

Chater II.

(12) S "being

the length
of void.

See

III

vo = muzzle velocity in feet

uo = total displacement up "bore in feet

pm = max. powder pressure, Ibs. per sq. in.

64'4 u

(15) mean powder
pressure,
Ibs.per sq.
in.

346

Ab » area of bore of gun.

If, ho*ever, the length of void corresponds to
• displacement greater than the recoil displacement
for the projectile to travel up the bore of tbe gun,
we have,

b 2u0 a0

t. - - (2.3 log — * — * 2) + t

a oo

or approx.

> (16)

3 uo Ai

5 T. * *•

where tt is obtained froa the solution of the
cubic equation,

C-|j ( -ji - - ) + Vfo } cos (0 * •) - x; - 0 (17)

n here

rfo

: V,

wv0 + 4700

2(Vf,-Vfo)

32.2

and X^

ao cos (0 + a)

(18)

27 a u
also P0b * 4" * (b+u)» d-12 PB V <19>

Powder reaction
on breech when shot leaves muzzle.

COI8TBAIS1D VILOOItT Of BBOQILi

(1) During powder pressure period.
Knowing R from the previous formulae, tbe con-
strained velocity of recoil nay be computed from the

347

free velocity curve as follows:

From equation (3) we have, V * V* cos (0+a) - —

«r

and the corresponding displacement

* Kt*

X » / ?f cos (D + a) dt - ~-

o 4mr

Kt*
» X* ees (0 + a)

X QH

smr

Thus we see the free velocity curve of recoil both
against time and displacement of free recoil is re-
quired in order to compute the constrained velocity
curve.

The free velocity curve during the powder period
is divided into two periods, (1°) the velocity of free
recoil while the shot travels up the bore, and (2°)
the velocity of free recoil during the expansion of the
powder gases after the shot has left the muzzle.

Lednc's formula gives us a means of computing (!•)
while Vallier'a hypothesis serves for the computation
of (2«).

From Lednc's formula, we have, during (1°) of the
powder period,

v - r^ — (20)

b + u

"b 2u u
t - 5 (2.3 log -5- +-B- + 2) (21)

where

a * travel up the bore in feet
QO a travel up the bore to muzzle

v * corresponding velocity of projectile in the

bore of the gun (feet per sec)
vo * muzzle velocity of projectile

t * corresponding time of the travel in seconds.

. t 27 Pm /I 27 Pm.a ~ .

*-..t(rB--»±/a-Ie-) -13

348

pm » max. powder reaction on base of projectile

B

wv0

pe 3 -••••• * mean reaction on base of pro-
0 jectile during travel up bore.

a 3 (b^o) 12

Farther from elementary dynamics, (see Chapter II)

(w+f)v

Vf • — (22)

wr

( *\
2

X a • i .»

r
or approx. ^

2
X, - (23)

where w > weight of projectile in Ibs.

v * weight of powder charge in Ibs.
wr > weight of recoiling mass in Ibs.

The procedure therefore, to compute the free
Telocity carve against time and displacement during
period (1°) is as follows

(a) Compute b and from it a,

(b) For various displacement up
the bore: compute v and t.
(Equation 20 and 21).

(c) Then from equations (22)
and (23), compute V and X .

Arrange the data in a table with corresponding values
of V , X and t.

349

Prom these values the constrained velocity carve during
(1°) nay be computed from equations (3) and (4). Front
Vallier's hypothesis, we have, during (2°) of the pow-
der period, for the total pressure on the breech
Ft, - Pob - C(t -t0) (Valuers' hypothesis)

where

C

t - t0 tt - t0 2(Vf, -Vfo)mp
hence

Now, from elementary dynamics, the change of momentum
along the axis of the bore, becomes,

t
/ Pfc dt = mr(Vf - Vfo) (25)

*o

Substituting (24) in (25) and integrating, we have

obo 4mr(Vf.-Vfo) r°

fying, we

have, for the free velocity of recoil,

0
Vf - Vfo » - (t-t0)(l -

The corresponding displacement of free recoil, along
the axis of the bore,

t
Xf * Xfo + / Vf dt (27)

*o

w
where w + -

Xf0 » - u0 uo = total travel up the
r bore in feet.

if if.*/* v{0dt^A,-t0><.t - 4..P(vt..Tfo) A*-*

350

Simplifying, the displacement of free recoil for tine

t, becomes,

The following initial values and constants are to be
substituted in equations (27) and (28).

fo " ~ — *fo

wv^ + 4700 w

t0 - (2.3 log * 4 2)

3 uo

» - — approximately.
3 v

27 B . , 27

Pm » aax. powder reaction on base of projectile.

a

wvo
^« * ft 7 ' J " * mean reaction on base of projectile

daring travel op bore.

27 t a
P0b " 7- b . .^' 1.13 Pffl « reaction on breech of

gun when the shot
leaves the muzzle.
The procedure, therefore, to compute the free

351

velocity curve against time and displacement daring
period (2°) is as follows:

(a) Compute Pm, Pe, and then b
and a as before.

(b) Compute Pob, to, (Vfi-Vfo and

Xfo.

(c) Then from arbitrary time intervals
between tt = T and to compute from
equations (26) and (28) Vf and Xf

Arrange the data in a continued table as in (1°) with
corresponding values of Vf, Xf and t.

Prom these values the constrained velocity curve
during (2°) may be computed from equations (3) and
(4).

MAXIMPM V1LOOITY OF COJ8TBAIIHD BIOOILt

The condition of maximum velocity of constrained
recoil is when the powder reaction exactly balances the
resistance to recoil, since before this condition the
recoiling mass is accelerated and immediately after it
is retarded.

Hence Pb cos (0 + a) - K = 0

t - t.

2.r(V -Vto) «

Hence the time at the maximum velocity of constrained
recoil, is obtained from either of the following
equations:-

t~t

Solving for t, we have

(30)

352

2m(Vf i-Vfo)[Pob cos(0+a) - K]

or t. * Pobt0 (30')

Pob cos A * a

Substituting tm in (26) and (28), we have,

Vfm - vfo + -«7 ^m-to) t1 ~ 4mr(Vft-vfo)
and

(32)

where Vfm and Xfm are the free velocity and displacement
corresponding to the maximum constrained velocity 'of
recoil.

BEOAPITDLATIQH Of FORMULAE FOB PRIBCIPLE PEBIQDS DURIKQ
PQITDER PRESanHE P1BTQD.

In the constrained velocity curve daring the
powder period, we have the following important
points:

(a) Velocity and displacement
of the recoil when the shot
leaves the muzzle.

(b) Maximum velocity and its
corresponding displacement of
recoil and time.

(c) Velocity time and corresponding
displacement at end of the powder
period.

Given data:

»r = wt. of recoiling parts.

VQ = nuzzle velocity.

w = weight of projectile.

w * weight of powder charge.

u * total travel of shot up bore.

353

Pm = max. powder reaction on "base of shot.

P^ = max. intensity of ponder pressure assumed

X area of bore,
b * length of recoil.

INTERIOR BALLISTIC OOMSTAHT8 BgflPIBED FOB VELOCITY CUBVB;

Pe * mean average powder reaction on base of shot *

2

!I°-

2gv0
B = twice abeissa of max. pressure,

27 Pm ,, /7 27

•re?--" i^-u

* max. velocity of free recoil

a velocity of free recoil - shot leaves muzzle
w + 0.5 w

pob * total pressure on breech when shot leaves
muzzle.

*

4 (B+u0)»
tQ * time of recoil while shot travels to muzzle

B , 2u v 3 uo
= - (2.3 i0g -- + - + 2) - - ~ approx.
a « vo

tt * time during the expansion of gases after
shot leaves muzzle.

m a(vf.-vfo) j^

pob <

354

t * T » time for total powder period

""""'*. * »,„ " " c"

Xfo » free movement of gun while shot travels to
muzzle

u0(w+0.5 I) w+0.5 w
» - = — — — u0 approx.

\f io = free movement of gun during expansion of
powder gases.

Total free movement of gun during powder period

B • *fo * xf'o

K » resistance to recoil: t * angle of elevation:
a * angle of plane of guides with horizontal,

» «• * f* \
- «rVfi cos (0+a)

b-(E-Vf ,T)cos(0+a)

VKLOCITY AMP PT 8PL AGKMR MT8 AT PERIODS

(a).(b) and fe).

At Period (a):

V0 and X0 » the constrained velocity and dis-
placement in recoil for period (a)
when shot leaves the muzzle.

Kt0
V0 » Vfo cos (0+a)

Kt«
X0 » Xfo cos (0+a) - ^—

355

At Period (b):

tm » time at max. velocity of constrained re-
coil.

K(T-t0)

Pobcos(0+a)

m and Xfm = velocity and displacement of free
recoil at the instant of maximum
velocity of constrained recoil.

pob ^ob^m"*©)

xfm - xfn * tVfo * « — (tm~to) ~ » — TTT-

10 2mr 6mr(Vf
V- and X_ = maximum constrained velocity and cor-

m in ••• *i^ ^ ^

responding displacement of recoil.
Ktm

x^ \ "I

Vm = Vfm cos (0 + a) - —

xm = xfm cos

At Period (c):

V = Vr = constrained velocity of recoil at end
of powder period.

Xt - EP a corresponding displacement of constrained
recoil at end of powder period.

Kt

V » V_ = Vf i cos (0+a)

mr

Kt2t

X, « EP = Xfi cos (0+a) »

oin_

f

356

UNITS TO BE EMPLOYED IN THE ABOVE AND FURTHER FORMULAE;

BRITISH 8Y8TIM MITRIO SYSTEM METRIC SYSTEM
QBAYITATIONAL 9R AV IT AT I 01 AL GRAVITATIONAL
UNITS. OMITS. UNITS.

Displacement in feet - ft. in meters*n in centimeters'

cm

Telocity in feet per in meters in centimeters
see. -ft/sec. per sec.> per see *
m/se c . em/se c .

Force

pounds - Ibs . Kilograms = Kilograms
kg. kg.

Pressure
Intensity

lb s . sq. in,

Kg. per sq.
cm.

Kg. per c q.
en.

Pressure
Area

. inches Sq.cn.

Sq. em.

Mass

Lbs/g (£-52.2) Kgs/g

Kgs/g
£ = 981

Ti»e

Seconds =Seo. Seeonds=8ec* Seconds - Sec.

OOJ8TRAH1D VKLOCITY COBVK:

(2) During Retardation Period of Recoil.

After the ponder period the recoiling mass is
brought to rest by the resistance to recoil. The
recoiling mass then reaches the extreme out of battery
position.

At the beginning of the 2° period of recoil, the
recoiling mass has an initial velocity Vt * Vri and
an initial displacement from battery Xt * Er.

357

A V
Prom the equation of motion, we have K » - m_V —

dX

Integrating, between the limits X, to any given
displacement X, and between corresponding velocity
V, to Vg we have

X Vx

/ K dX - - mr / V dV (33)

X V.

Hence,

retardation period of the recoil. Hence

K(X-X )» - which is the equation of
2

energy during the

2K(X-X. )

(34)

rar

A simpler and more direct form for computing the
constrained velocity during the 2* period of recoil is
as follows:

We have, as before K dx * - mr V d V

Integrating between the limits X and b in the
displacement and Vx and o in the velocity, we have

m.Vx
K(b-X) - -J-^- (35)

/2K
— (bH

Hence Vx - / — (b-X) (36)

ror

showing that the velocity during the retardation
period is a parabolic function of the displacement.

It is to be especially noted that a characteristic

of a constant resistance to recoil is a parabolic

function of velocity against displacement.

GENERAL EQUATIONS OP RECOIL In the previous formulae
CONTINUED.- VARIABLE RESIST- the resistance to recoil
ANCE TO RECOIL. was assumed constant

throughout the recoil.

It is however often de-
sirable for stability to decrease the resistance to re-

358

coil in tbe out of battery position and thus partially
compensate for tbe decreased stability due to the
moment effect caused by tbe overhang of the recoiling
mass in the out of battery position.

With a variable resistance to recoil it is customary
to maintain a constant resistance during tbe powder
period and thence decrease tbe resistance proportional
to tbe displacement to the out of battery position,
with a given arbitrary slope "m". See Chapter III.

Let KQ * the constant resistance during the pow-
der period.

Vt and Vr * tbe velocity of constrained recoil

at the end of the powder period,
b = total length of recoil.

Then tbe equation of tbe resistance to recoil
against displacement of recoil becomes,
K0 » constant, from 0 to Xx or Br

(37)
K * Ko — m(X-X ) from X^ to

Further, g j

Vt » Vp » Vf • cos(0+a) 2. (38)

mr

K0T«

X * Er » B cos (0+a) - (39)

o rn —

Now fron tbe equation of motion of the recoiling parts
during the retardation, we have, K dX » - mr V dy

Integrating between limits, X, and b: and V, and 0,

K dX

d V

Hence ^ m y*

[I0 - m(X-Xt)J dX > -j-i

Integrating, we have for the energy equation,

359

K0(b-Xt)

»(b-Xt) mrVi

(40)

Substituting (38) and (39) in (40) and neglecting the

terra j/*T4
m ^o1-

in the expansion as small, we have

ra "oj
2 4

f»rVf.
< -

2

cos*(0+a)+ ^ [-b-

]
>

, T«
b-B cos(0+a)+ Vf cos(0+a) T - - — [b-E cos(0+a)]

o ffl—

Thus from the ballistic constants E and T, together
with the length of recoil, maximum free velocity of
recoil and any given arbitrary slope "»", the re-
sistance to recoil maintained constant during the pow-
der period may be computed.

Substituting KQ in place of R in the proceeding
formulae during the powder pressure period enables as
to compute the retarded velocity curve daring the
powder pressure period.

During the retardation or second period of recoil
we have, K dx » - mrV dV

Integrating, from the displacement x to the end
of recoil, we have
b v
/ Kdx * mr / J Y dV

therefore / (Ko ~ «(X-Xt)]dX »

x

Hence

mPV*

K(b-X) -

•X

mX,X

and simplifying, we bave

360

"rVl

[K0 - - (b+X^X, '
2

Hence j =

:(K - - tb+X - 2X)](b-X)

O

(42)

where as before,

m » the arbitrary slope of resistance to recoil.

o m*

X a E cos ()0 + a) - iv- —
*mr

GENERAL EQUATIONS OP When the direction of recoil
RECOIL - Cont. is along the axis of the bore,

(a) Constant resistance to
recoil throughout recoil,

let K » B + R - Wr sin 0 = total resistance to recoil
B * total braking R = total friction

E 3 displacement in free recoil during powder

period.

T = corresponding time for free recoil,
then for the motion of the recoiling parts,

dV T pb KT

Pb - ' mrdT ( ~f dt * m7 3l Vr

but as before / ~ dt = Vf = max. free velocity of

recoil, hence

KT*
and the corresponding displacement, X *Er » E —

0IH«B

After the powder period, from the equation of
energy,

—iH-V.. » K(b — X.)

KT

* / .. Ri.a .. /. Rlv

- m_(Vf ) ' K(b - E * - — )

m P c ffl j.

361

t Tf«

*"r*f
and simplifying, we have K » (43)

This equation obviously is a special case of equation
(7) since when (0 + a) * 0, cos(0 + a) = 1 and a = - 0,

(b) Variable resistance to recoil.

The resistance to recoil as before is assumed
constant during the powder pressure period and thence
to decrease uniformly consistent v/ith stability, that
is with a stability slope as given in Chapter III on
stability.

At the end of the powder period, we have for the
constrained velocity of recoil and corresponding dis-
placement,

At the end of recoil, the resistance to recoil be-
comes KQ - mfb-Sr) where m * the stability slope
(See Chapter III).

The mean resistance from the end of the powder
period to the end of recoil, becomes,
2K-m(b-Er)

- — = K0 - ? <b-*r>

2

and from the equation of energy of the recoiling
mass, we have

K00>-Br) - 2 (b-Ep)* - 7 mr V* (46)

Substituting the values of Er and Vr from (44) in

m

2

(46) and neglecting the term _*_»

m K T

we have,

mV * m(b-B)a

2 -r

362

This equation obviously is a special case
of equation (41) since when (£J + d) » 0, cos(j0+a) * 1
and d » - 0. .*..

(c) Dynamic equation of recoil
during powder period.

Since during the ponder pressure period, the re-
sistance to recoil is assumed constant even with variable
recoil, we have, therefore, the same dynamic equation
with either variable or constant resistance to recoil
during the powder period.

Dividing the powder period into two intervals tQ
and tx - to while the shot travels up the bore and
during the expansion of the powder gases after the shot
has left the bore, respectively, we have

(1) During the travel up the "bore,

/ ^

(48)

Kt

u - — (49)

8 2u u
and t - -(2.3 log — + - + 2) (50)

a B B

Thus V, X and t are functions of the parameter u. The
ballistic constants a and B have been determined
previously in this Chapter as well as in Chapter III
in "Interior Ballistics".

363

When the shot reaches the muzzle,

/ \

("*")

2

Kt

(B + u0) mp

(61)

w

(52)

o o
and t - - (2.3 log — + — + 2 )

3 uo
* — — approx

^ o

(53)

(2) during the expansion of the powder
gases, we have

au

uo where u = the total

travel up

the "bore, the dynamic equation of recoil during this
period becomes,

ir a dV
- K » rar -

dt

(54)

2

where Pob » B2 -- - - 1>12 pm (See Chapter III)

4 \B + u - /

Integrating, we find

(55)

l "O-

364

Hence V - V.+ - - - [1 -- - ] _ _ (t-t0)(56)

2(tt-t0)

The corresponding displacement is obtained by integrat-
ing equation (55)

2 6(t-tQ)

V d(t-t)

+ mr V0 (t-t0) + Const. (57)

Now mr / V d(t-t0) = m(X -XQ). Hence where t = tQ,
X » X0 and const. * 0. Simplifying (57) we obtain

for the recoil displacement during the second period
of the powder period,

X = X0 + V0(t-t0)

2m

(58)

To obtain the maximum restrained recoil velocity and
corresponding displacement, we must equal the total
powder reaction to the total resistance to recoil,
that is Pb - K » 0

Pobd -- ) - K » 0 where PO^ » the pressure on
\~t-o tnc Creech when

the shot leaves
the muzzle.

t = total time to maximum restrained recoil
velocity, hence solving for tm, we have

•
ta » tt - — — (tt - to ). Substituting in equation

ob (56) and (58) we have

Ppb^nrV . tm"to . K

2(tt-tQ) m

365

(t -t )* P°b(tm'to) M- *•"*»
* 3

2^ *<tt-t0)

(60)

At the end of the powder period,
t » tt = T and X » Er and V = \ and substituting

again in eq.(5c) and (58), we have

(t -tn ) K

Vr - V0 * Pob -i- -- ~ (tt -t0 ) (61)

2mw fflr

>*

(62)

(d) Dynamic equation of recoil
during the retardation or the
pure recoil period.

(1) constant resistance to recoil:

Since the total resistance to recoil is constant,
the velocity must be a parabolic function of the
displacement of the recoil,

Prom the principle of energy, we have,

M V* /~~~t \~

K(b-X) » — — hence V » /2

2 mr

(2) Variable resistance to recoil
The resistance to recoil out of battery, becomes,

k * K - m (b-E + - — ) where K =
2mr

• "~*~

The average resistance to recoil in the displacement
b - X, becomes

k * \ (b-X)
From the energy equation, we have,

366

k(b-X)+ I (b-X)' - j-j»r V*

I
2

/Kb-XHTc* • (b-X)]

v

COMPONENT REACTIONS OF Let K * total resistance
THE RESISTANCE TO RECOIL. to recoil. (Ibs. or

Kg)
B = total braking. (Ibs.

or Kg)
R » total friction to

recoil. (Ibs. or Kg)

Ph = reaction of hydraulic brake. (Ibs. or Kg)
Pv * reaction of recuperator. (Ibs. or Kg)
px * hydraulic brake pressure. (Ibs/sq. in) or (Kg/I2
A 3 effective area of hydraulic brake piston, (sq.in.

or m )

py » recuperator pressure. (Ibs/sq.in) or (Kg/m )
Ay * effective area recuperator piston. (sq.in or m )
Vo " initial volume of recuperator. (cu. ft. or m )
X * recoil displacement. (ft. or m)
Sf * final spring reaction. (Ibs) or (Kgs)
So = initial spring reaction. (Ibs) or (Kgs)

The total resistance to recoil then becomes
along the bore along special guides

K * B + R - Wr »in 0 K»B+R+Wr sin 6

where 0 » angle of elevation, 9 * angle of guides.

Now in systems where the hydraulic brake is independent

of the recuperator system, B » Pn + Py

In systens where the brake and recuperator are
connected B » Pj,

For independent systems

PVJ(T T~\~] ^or pneumatic recuperators

""

367

Sf-so
v » S0 + ( ) x for metallic recuperators

o

and Pyi» 1.3 Wr(sin 0, + u cos 0m) approx.

hence

KAS
-7- where c = -7-

wx "x

V k *
vo * cv

V —A*

ro

) * ~r~ for pneumatic re-

"v

cuperators.

sf-so cv*
B = S0 + ( ) x + — 7— f or metallic re-

x cuperators.

For systems where the hydraulic brake and recuperator
are directly connected,

, KA'

P-PV = — — where c = ——

* Mf* M*

wx "x

pviA * Pyj * 1.3 Kr(sin 0 + u cos 0) approx.

therefore
P =

• »
o v

pA

' i *
c AT

since

and c A » C hence B

cv

~

which is an equation of exactly the sane form as for a
system where the recuperator is independent of the
hydraulic brake.

368

The general equation for tbe resistance to recoil
"becomes,

(a) when the recoil is along axis of tbe
bore:

a

cv
* ~ — + R - Wr sin t, for pneumatic

V

recuperators.

K - S0 + ( — - - )x + -yr + R - Wr sin 0, for metallic

recuperators,
(b) when tbe recoil is along special
guides:

V *

O CV

K = pvi(VTT4 — ) * ~T~ * fi * wr sin e» for pneumatic

V f^ AX W y

recuperators.

SfSo

+ -7- + B + Wr sin 6, for metallic

recuperators.

a
CV

K » -7- + R + ffp sin 9, for gravity mounts.

GENERAL EQUATIONS OP The function of the recuperator
COUNTER RECOIL. is to return tbe recoiling mass

into "battery. The stability of
a mount in counter recoil is
greatest at the beginning of

counter recoil and least at the end of counter recoil
or when the gun enters tbe battery position. To
prevent shock and unstableness as the gun arrives in
battery it is necessary to introduce some form of
counter recoil buffer towards tbe end of counter re-
coil. Very often a buffer resistance of varying
amount is introduced throughout the counter recoil.
In addition we always have the resistance of tbe
guides.

Without a recuperator tbe recoiling mass must be

369

returned to battery by the gravity component due to
the inclination of the guides with the horizontal.
If this inclination is small, the gravity component
does not greatly exceed the friction and thence a very
elementary buffer may be used, the return velocity
being always small.

Let KV 3 total unbalanced force in counter recoil.

Fy = recuperator reaction.

= variable orifice for counter recoil buffer.

By = counter recoil buffer resistance.

Ay = effective area of recuperator piston
py = pressure intensity in the recuperator
cylinder.

pa = pressure intensity in the air reservoir.
R = total friction of counter recoil.

During the accelerating period of counter recoil,

we have

dv

Kw = HD v — and during the retardation
dx

dv

Kv » - mR v — -

dx

During the acceleration Kv is necessarily always
smaller than the total resistance to recoil, "hence
during the acceleration counter recoil stability is
of no consequence. During the retardation, if

d1 = the distance from front hinge or wheel contact
with ground in a field mount, to the line of
action of the total resistance to recoil.

L = horizontal distance between front and rear

supports of mount.
Ls = horizontal distance from rear support to

center of gravity of total system with

recoil parts in battery.

b = total length of recoil.

370

*3 = weight of total mount.

Then, for a gun recoiling along the axis of the bore
during the retardation, Kvd' ^ ltg(L-Ls) +Hr(b-X)cos 0

and the minimum stability occurs when the gun enters
"battery, that is Kyd ' ^ WS(L-LS). The stability slope
of counter recoil, becomes ^ cos ^

m1 * — — — —
d1

To consider the components of the total resistance to
counter recoil, we have three classifications:

(1) recuperator systems independent of
the hydraulic bralce and with no throttling
between the air and recuperator cylinders.

(2) recuperator systems independent of
the hydraulic brake, with throttling
between the air and recuperator cylinders.

(3) recuperator cylinders connected
directly with the brake cylinder. In all
systems an independent buffer may be in-
troduced in either the recuperator or
brake cylinder front end. In certain
types the buffer acts as a plunger brake
within the piston rod of the recoil
brake.

Then,

(1) for recuperators independent of the "bralce
cylinder and with no throttling between ths
air and recuperator cylinders,

Kv = Fv - B'X - Wr sin 0 - R (1)
when

0

Vo = initial volume

1.3 Wr'(sin 0+0.3 co« 0) approx.

(2) for recuperators independent of the
bralce cylinder, with throttling between
the air and recuperator cylinders,

371

pvAy -BjJ - Wr sin 0 - R

where « 2
c v

pv " pa 2 (WQ = constant orifice usually)
wo

V^

Fyj. = 1.3 Wr(sin 0 + u cos 0)

hence

• 2
c v i

KV =(Pa 2~'Av ~ Bx " wr sin & ~

and since » 2

"o

wo

then the equation reduces to same form as (1), that
is Ky = Fv -(Bi + BJ ) - Wr sin 0 - R,

(3) for recuperators directly connected with
the recoil bralce cylinder,
Ky * pyA - B£ - Wr sin 0 - R

where » »
c v

P\r = Pa W2 ("x = variable orifice

by buffer rod on a
floating piston in
recuperator or air
cylinder.)

rt« ^V0-A(b-x) J - Fv
F . = 1.3 Wr(sin 0 * u cos 0) hence,

Ktt = F - (B* + B") -Wr sin 0 - R,
v v

" 2
« C V

where Bx = ~ — Av

A ,,7 V

"x

which is again an equation of same form as (1).

372

The general equation of counter recoil, therefore,
becomes

*V ~(&x + Bj) - Wr sin £J - fl » mg v -—

where

9 t

a

i

L/n Y *

| y

Bx

" c ~
»x

2gc"w»*

o"

DAjv*

»

* V

Bx

» * " *

1 c .«

2gC w „

«•»

CALCULATION OP RECOIL It is often convenient to
CURVES. calculate the retarded velocity

curve against displacement,
especially when the resistance
to recoil is not made constant.

In all cases we have seen the resistanc-e to recoil is
in general a function of both the displacement and
velocity of recoil, that is the recuperator component
of the recoil resistance is a function of the displace-
ment, whereas the bralce component is a function of the
velocity and the variation of the throttling orifice.
Hence K = f(xtv) and the dynamic equation of recoil

18 dV

Pb cos(0 + 0) - K » WR — - or when the recoil

translates in the

direction of the axis of the bore,

K dV

To measure Pj, we may consider the momentum im-
parted by the powder gases in free recoil, then

Pb " «R aT

or J Pfcdt » BR^f"^ft) Therefore, for

li the same interval

of time (t-tt) we have

373

»R(Vf-Vfi) cos (9*0) - K (t-tt) « «R(V-Vt)

be nee

V » Vt+(Vf-Vft)cos(9+0) U-tt) or when the re-

"R coil translates
in the direction of the axis of the bore,

R *

V = V +(Vf-Vf.) (»-t.) Further since X = X +/ Vdt,

Bo •

t

we have

t
X = Xt*Vt(t-tt)+ / Vfdt cos (9+0)-Vft(t-tt)cos(9+^) -

K (t.t )* now J Vfdt = Xf-Xft hence
2«R 4t

X = Xt + [t-Vf

t
(t-t ) or when the recoil translates in the

direction of the axis of the "bore,

x - x^O^-v^Mt-^MXf-x^)- Jj- (tf-ta)*

Therefore the velocity and displacement, for any given
interval (tt-ts)

(a) along guides not parallel to

the "bore:

vt»vt+(vft-vfl)cos(e+0) - — (tt-tt)

•R

xt=xt+(?t-vftcos(e+0)](tt-ttMxft-

(t,-tt)«

(b) alon^ guides parallel to the
axis of the "bore:

374

After the powder period these formulas reduce to

w -".-'t)

•R

2nR

and obviously are independent of the direction of the

guides with respect to the axis of the tore.

Kt +K2
The value of K = — - , which may he closely

approximated by a

repetition of the substitution in these equations,
since from the first substitution we closely ap-
proximate V2 and thereby can determine Ka=f(XaV )
for the second substitution.

CALCULATION 0? ACCELERATION, TIME AND DISPLACEMENT
PROM A GIVER VELOCITY CURVE:

Recoil and counter recoil

velocity curves are usually drawn experimentally as
functions of the displacement though they may be
drawn as well as functions of the time. The customary
method of obtaining a velocity curve, is to set a
tuning fork vibrating and allow the vertical oscillations
to form a sinuous curve along a narrow soot covered
strip recoiling with the gun. Then if f * the fre-
quency of oscillations of the fork, we have for the
time of one oscillation, T = -^- If n = the number of

oscillations for an
interval Ax, the velocity becomes,

v 3 — , where At = nT if x is measured in inches,
At

' ' 13 ZT (n/sec)

375

To obtain the time as a function of the displace-
ment, since vdt=dx

t = / - dx

o v

1 * 1

and if x is measured in inches, t = — / - dx

v o v

Hence the area under the reciprocal of the velocity
curve against displacement is the time of recoil.

We may then draw the velocity curve as a direct
function of the time of recoil.

When the recoil velocity is measured as a function
of the time, the acceleration is

dv

•7- = the slope of the velocity curve

at

When the recoil velocity is measured as a function
of the displacement, the acceleration is,

dv

v ~ = the velocity * the slope of the velocity

curve .

= t~he sub-normal of the velocity curve.
If dx is measured in inches, the acceleration is

12 v — (ft/sec*)

dx

From the relations, v=f(x) and t=/ - dx =/ — - — - dx

v f(x)

we see that the velocity curve may "be readily expressed
either as a function of the displacement or as a function
of the time or both.

CHARACTERISTICS OF RECOIL From Proof Firing T«sts,

CURVES. recoil curves are obtained

for both recoil and counter
recoil. From these curves,
it is possible to determine

the variation of the reactions throughout recoil or

counter recoil.

378

In the analysis of curves during the powder
period, since the mutual relation connecting the
variation of powder force and the retarded recoil is the
common time, it is necessary to express the forces,
velocities and displacements as functions of the time.

In the analysis of curves during t~he retarded
recoil and counter recoil it is possible to express
the forces and velocities as direct functions of the
displacements which considerably simplifies the work.

(1) Powder Pressure Period: Recoil along
axis of "bore. The equation of recoil is

dV

Pjj - K = Dp T— where K = B+R-Wrsin 0

With a given velocity curve, the velocity and displace-
•ent should be tabulated as a function of the time;
then for any interval (ta~tt) "e have

(vft-vfi>- <Vvt)-L(tt-tt) = o

= o

If K is assumed constant or found to be constant by
brake measurements or if it is determinate as a
function of the displacement, we nay evaluate Vf the
free velocity of recoil. More often however, the free
velocity and displacement curves can be evaluated as a
function of the time, and knowing the retarded velocity
and displacement curve as a function of the time we
may calculate the resistance to recoil from the above
expressions. Then u _y

pb = "R /
la - *t

and ,4? dV dV

pb-»R jf '- *here "R d"t s "R v al ' "R0

dV
It is to be noted that Pv and - Dp r— are the external

d t — — ^^^—

377

recoil forces during the powder period. Further P^ acts
along the axis of the bore and - nR £*- acts through

the center of gravity of the recoiling parts parallel
to the axis of the bore or guides. If e = the distance

from the center of gravity of the recoiling parts to
the axis of the "bore, we have for the external reactions
on the mount a couple Pue and a force parallel to

dV

the axis of the "bore, Pv - mo — — = K. The balancing

dt

forces are the weights and reactions of the supports.
For stability the moment of the weights about the
rear support must exceed the moment of Pve and K
about the rear support.

(2) Retardation Period: Recoil along
axis of bore. During this period, we
have simply

dV

dt

applied through the
center of gravity of the

recoiling parts,
«R V — = - K parallel to the axis

of the bore,

which together with the weights and balancing support
reactions are the external forces on the mount.

It is to "be further noted that since

X * Ph * Pv * R ~ wr sin ^ we nave

velocity curve,
dV
Ph » - oR V Pv - R + Wr sin 0

f dx

V k

where Pv * ?vi I — ) f°r pneumatic recuperators

V - A
vo *x

R = 0.25 lfr cos 0 + R_ approximately where Rp = estimated

packing friction.

378

(3) Counter recoil: C'Recoil along axis

of bore.

During the accelerating period of counter recoil, the
inertia resistance is directed towards the breech the

same as in recoil. Here

dv

Kv = mp v — — to t"he rear
dx

and during the retardation the inertia resistance is
directed forward and "here,

*v - * »R V £

which together with the weight of the system and
balancing supporting reactions are the external
forces on the mount.

Since further, during the retardation,

dv i
- mR v — = Fv - Bx - Wr sin 0 - R we have

sin

and v

o »
Fv * Fvf [ ] fof pneumatic recuperators

Vft-A(b-x)

and

R 3 0.15 Wr cos 0 + Rp approximately where Rp =

estimated packing friction.

Since critical counter recoil stability is at
horizontal elevation, C'recoil curves are usually ob-
tained at "horizontal elevation. Then,

i dv

Bx»?v-R+»pV— for the buffer force where

^x the overturning force is
dv

•D v -— along the axis of the bore forward,
dx

RECOIL BBAKII0 WITH A CONSTANT ORIFICE:

As a first approximation we will assume tne re-
cuperator reaction not to vary greatly in the recoil.

379

Then K = A * Bv where A = Pv + R - Wr sin t

B = the hydraulic "brake
throttling constant.

(1) During the powder period, we
have

Pu - (A+BV*)»mR —
dt

V +V • .

mR
Expanding, we have

which is a quadratic equation of the form
aV* * bV + c = 0

2 Z

and

- b ± A) -4ac
V, • — where a . ^(t^J

V B
b » 1 * -i- (t,-tt)

2 nn

c =

,-t -t-

4«R "R

If the intervals are talcen very small, then

A +BV*

(t,-tt)*(vfl-vft)

Then solving for V2 we may repeat with the expression
V +V

V, = V± -- - (t.-tt)*(Vf,-7ft)

380

for a closer approximation.

The displacement is obtained from the expression,

V +V

(2) During the retardation, we have
V — » - A - BV* hence dx

•R v — a ' A+BV«

•R A*Bv

Integrating, we have X -X *— loge

28 '

In particular if Xt * Eg the constrained recoil dis-
placeaent at the end of the powder period and vi = Vp

tbe constrained recoil velocity at the end of the
powder period, then, for any displacement x and recoil
velocity V, we have,
»R ,

x " ER " 7* loge

or with common logarithm,

X - ED « 1.15 — log -
B A+BV»

when V = 0 the length of recoil "becomes,

b - BR + 1.15 — log (1+ - Vj)

B A

As • first approximation, we may take Eg * E the
displacement in free recoil during tbe powder period
and VR * Vf the maximum velocity of recoil* then

h - B * 1.15 -| log (1* j-

CO

CHAPTER VII

CLASSIFICATION AND CHARACTERISTICS OP RECOIL AHD RE-
CUPERATOR SYSTEMS.

Recuperator systems nay be divided into:

(1) Hydro pneumatic recuperator systems

(2) Pneumatic recuperator systems

(3) Spring return recuperator systems.

(1) With hydro pneumatic systems, we have two

fundamental arrangements:-

(a) The hydraulic brake separate
from the hydro pneumatic re-
cuperator. This requires two or
more rods, a brake rod and a re-
cuperator rod. Further we have
in general two or more cylinders,
a brake cylinder, a recuperator
cylinder which may have passage way
or connection with an air cylinder.
The recuperator and part of the
air cylinder is filled with oil.
The oil nay be in direct contact
with the air in the air cylinder as
in the Schneider and Vickers
material or it may be separated
from the air by means of a float-
ing piston in the cylinder.

(b) The hydraulic brake cylinder
connecting directly with the recuper-
ator cylinder. The oil must be
throttled between the recoil and re-
cuperator cylinder, and thus the oil
at lower pressure reacts usually oa

381

382

a floating piston separating the
oil and air in the recuperator
cylinder. To augment the initial
volume for the air in the re-
cuperator an additional air
cylinder connecting with the re-
cuperator may be introduced. Thus
with this arrangement we have from
two to three cylinders.

(2) With pneumatic recoil systems, we have usually,

(a) One or more pneumatic cylinders,

according to a satisfactory layout.
The piston compresses the air directly, no oil
or other liquid being used for transmitting the pres-
sure.

(3) With a spring return system, we may have
various arrangements:

(a) One or more spring cylinders
separate from the recoil brake
cylinder.

(b) With small guns, the spring con-
centric and around the recoil
brake cylinder.

The potential energy or the energy of compression
of the recuperator during the recoil, becomes

?OR PNEUMATIC OR HYDRO PKBUMATIC 3Y3TIM8;
If

paj - initial air pressure. (Ibs/sq.in)
paf = final air pressure (Ibs/sq.in)

Paf

— — = m =» ratio of compression

Pai

VQ » initial air volume

Vf * final air volume

Ky * recuperator reaction

383

Un

T\\\\\\V? \\\\\\\\

Z1C

K\\\\\\V>

oo

ul

VA'.W

384

Vf k »f

/ Pa* » - - Pai »5 /

1-k V, V0

Paf ?o k

Now m » - 3 <.r~) 3 the ratio of compression
Pai vf

Therefore, the work of compression becomes in terms
of m, and the initial volume,

I»(p) - 1] ft. Ibs. (1)
where pas is in Ibs. per sq. ft. and VQ in cu. ft.

[.(, -u ft. lba. (1.,

12 k-1 k

when VQ is in cu. inches and pa^ in Ibs. per sq. in.
The recuperator reaction, becomes for any displacement
X in the recoil,

Pa *a « Pai

V0 k

-A .

o flax

where pai is in Ibs. per sq. in. and Aa in sq. in.
Jf in inches and VQ in cu. inches. The initial volur
becomes,

v0 • V ^7- (s)

where b = length of recoil. With the oil in direct
contact with the air, we will assume that the

385

temperature remains approximately constant through-
out the recoil and k Mill be taken at 1.1

With a floating piston interposed between the
oil and air, or with a pneumatic recoil system, we
will assume a negligible radiation, that is the com-
pression approaching an adiabatic condition.

Hence k will be assumed = 1.3

FOR SPBIH3 RITUHH SYSTEMS:

If

So * initial spring recuperator reaction
Sf * final spring recuperator reaction

Then the potential energy stored in the spring at the
end of recoil, becomes

b Sf-Sft

P,g, . / (* + _L_2.x)<ix.

•<VSf>f (4)

b"
and if b is inches, we have P,E,*(So+Sf )—-

mm

The reaction exerted by the springs at any displace-
ment of the recoil X, becomes

sf-s0

Kv * So + — r x (5 )

RECOIL BRAKES. In the broad classification
of recoil brake systems, we have
those: (a) where the brake system
is independent of the recuperator
system, (b) where the brake system

is part of or connects with the recuperator system.

(a) In consideration of independent
brake systems, we nave a further

386

classification-
CD Brakes with throttling orifice
through the recoil piston, the vary-
ing aperture during the recoil being
produced by the difference in areas of the
constant apertures in the piston and the
area of the bar or rod of varying depth
or diameter fixed to the recoil cylinder
and moving through the aperture; or the
throttling nay be around the piston by
varying grooves in the cylinder walls
along the cylinder.

(2) Brakes with varying apertures through
the recoil piston, the aperture being
cut off during the recoil by a rotating
disk about the axis of the piston, the
disk being rotated during the recoil by a
projecting "toe" engaging in a helicoidal
groove in the cylinder wall. This form
of brake is known as the Krupp valve and
is extensively used not only by the
Krupp but other types as well.

(3) Brakes with the throttling taking
place around the piston, [not through
as in (1) and (2)], through a sleeve
perforated with boles along the recoil.
The piston cuts off the number of boles
during the recoil thus decreasing the

effective throttling area.

(4) Brakes with the throttling taking
place through a spring controlled valve.
With independent brake systems the spring
valve is contained in the piston. Where
the brake is part of the recuperator the
throttling takes place through a fixed
orifice sonewhere between the two cylinders

(5) Brakes with a constant orifice. On-

387

less the air pressure is fairly large,
and the throttling relatively small
constant orifice control should be avoided
since it gives a large peak in the
braking.

In consideration of brake systems as a part of the
recuperator, the throttling takes place between an orifice
fixed somewhere between the two cylinders, and usually
of the spring controlled type though sometimes with
high air pressures a constant orifice may be used.

In general it nay be stated when the recoil energy
is large the throttling may be very satisfactorily con-
trolled, as in brake systems of the type (1),(2) and
(3). Where the energy of recoil is small as in small
caliber guns, the throttling areas especially at the
end of recoil must be small. This can not be satisfact-
orily met with "bars" or "grooves" due to the inherent
tolerance making very often the clearance greater
than the required throttling areas towards the end of
recoil. This difficulty has been repeatedly met in the
design of small recoil systems. On the other hand
spring controlled valves are admirably adopted for
small recoil systems, since the throttling towards the
end of recoil can be finely controlled.

COUNTER RECOIL SYSTEMS OR HUNHIKJ 70BWARB BRAKES:

In the classifications of counter recoil systems,
we have two general types of systems:

(1) Where the brake comes into action
daring the latter part of the counter
recoil.

(2) Where the brake is effective
throughout the counter recoil.

With (1), the buffer action can only take place
after a displacement of the void (the displacement

388

of tha recoil piston rod » Ar « b), which with guns
of large piston rods May be a considerable part of
the counter recoil.

With (2) the buffer must be filled daring the
recoil, otherwise no buffer or braking action can
take place.

The brake with systems where the buffer action
takes place towards the end of counter recoil, con-
sists usually of a buffer chamber as an extension of the
recoil cylinder in the front and spear buffer pro-
jecting from the front side of the piston or with a
buffer chamber within the piston rod itself the spear
buffer rod being attached to the front bead of the
cylinder. In the former type we must have a projectory
chamber from the cylinder, while in the latter we must
have a larger piston rod with consequent larger void
to overcome during the counter recoil.

With guns of high elevation in order to meet
the demands of counter recoil at maximum elevation,
we have a surplus potential recuperator energy in
the recuperator and no means of checking or regulating
the velocity during the greater part of horizontal recoil:
therefore at the initial condition of counter recoil
stability, we have unfortunately an inherent con-
dition of a large buffer force, making the mount un-
stable at the end of counter recoil.

Therefore, this type of counter recoil brake,

which is effective only during the latter part of
counter recoil should only be used for guns of low ele-
vation.

Counter recoil brakes of type (2) which fill
during the recoil end are effective throughout the
counter recoil, are really the standard form of
counter recoil regulator to meet the varying con-
ditions required in modern artillery. Varying forms
of this type of brake are used. Thus in the Filloux
and Schneider reeoil system the buffer is at the end

389

of a counter recoil rod which serves also as a
throttling bar through the recoil piston. The
buffer head is enclosed and slides within a buffer
chamber in the piston rod. Apertures near the piston
in the piston rod adait the oil daring the recoil into
the buffer chanber, the oil passing through a valve
in the buffer bead which can open during the recoil
and closes during the counter recoil as in the
Schneider material. In the Filloux, though we have
a filling in buffer in the recoil piston rod, the buff-
ing action takes place only at the end of counter re-
coil but a positive filling is ensured. The velocity
of counter recoil is maintained low in this system
by lowering the recuperator pressure during the
greater part of counter recoil by throttling through
a constant orifice in tbe air cylinder.

Various forms of filling in buffers are shown
in figs. (1), (2), (4).

APPROXIMATE FORMULA FOR If tbe total resistance
TOTAL RESISTANCE TO to recoil is assumed con-
RECOIL. stant throughout the re-

coil, we have when the
recoil is along the axis
of the bore, which usually occurs in practice, that

t ..a
7 "r Vf

b-B+?fT
where

B * free reeoil displacement during powder

period.
T * tine of powder period.

wv + 4700 w
V « • * max. free velocity of recoil.

"r (ft/sec)

b * length of recoil, (ft)

Now

B * Ct VfT and T * Ct —

390

where

uo = travel up the bore and vo * muzzle velocity.

Substituting,

f
v v

— uo * c ~~ uo

This value of E may be obtained in another way,

•

"T v,

B = C( - )u0 - C — u
r

hence

0
"

1

K » 7 mr vf< * >

vf °o

b-C C — u0+C Vf —

vo vo

- 7 -r vf<

uovf

b*(C,-C C ) -
vo

Mr. C. Bethel found from computation on a series
of guns of various calibers that the value (Cj-C1C )
eould be represented very closely to a linear function
of the diameter of the bore, that is

Cf-CtCf

where

d = diameter of the bore, (in)
If

uo» travel up the bore (in)

?o= nuzzle velocity (ft/sec)

Vf » velocity of free recoil (ft/sec)

b * length of recoil (ft)

then

Cf- CtCt » .096 + . 0003 d

391

and we bare, K

2 uovf

[b+(.096+.0003 d)

(Bethels Formula)

The formula applies to a constant resistance to recoil
and is especially useful, since the computation of E
and T are not needed.

GENERAL EQUATIONS OP The characteristics and
RECOIL AND COUNTER functioning of the various re-
RECOIL.- RECOIL coil systems may be shown in
SYSTEMS. an unique way by a study of the

equations of recoil and counter
recoil. Let K = the total resistance to recoil assumed

constant throughout the recoil, (in Ibs)

Pb = powder pressure on breech

p * the pressure in the recoil brake cylinder.
(Ibs/sq.in)

A = the effective area of the brake piston. (sq. in)
py « the recuperator pressure (Ibs/sq.in)

AT » the effective area of the recuperator pis-

ton (sq.in)

B = pA + Pv*v s tne total braking, (in Ibs)
R_ * the total packing frictions, (in Ibs)
Rg * the total guide friction (in Ibs)

R * Rp+Rg»the total recoil friction (in Ibs)

0 » angle of elevation of the gun.

X = displacement from battery along the recoil

(in ft)

b = total length of recoil (ft)
Then during the recoil

dv
Pjj-K * mr — during the acceleration

- K - m_v T- during the retardation.

* at

392

The external force on the mount during the
acceleration is

dv

Pjj - «r — * K, as well as the weight of the

dt recoiling parts, and a couple

P^d^, where d^ = distance from
the center of gravity of recoiling parts to the

* *U V

axis of the bore.

During the retardation, the external force on the
mount in the duration of recoil is,

dv
— »pv — * K , together with the recoiling

dx weight,

(1) when the brake system is independent

of the recuperator system, then
K » B * R - wr sin 0

» pA+pvAv+R-wrsin 18
Now the hydraulic pull becomes,

C v*
pA -

•5

where

v * the velocity of recoil at displacement x (ft/sec)

Nx - the throttling are at displacement x.

Further, with pneumatic or hydropneumatic re-
cuperators,

k
PVAV ' Pvi

) where k = 1.1 to 1.3

and with spring return recuperators
PTAT - S - S0 * -~— - x

Hence the total resistance to recoil, becomes, with
pneumatic recuperators,

V,

393

and with spring return recuperators

* <?— ^
cv af so
K - — + (S0 + — £ — x)+R-Wrsin 0

Thus we have four components in the total resistance
to recoil,

(a) The hydraulic throttling component
which varies as the square of the
velocity,

(b ) The elastic reaction which in-
creases as a function of the dis-
placement.

(c) The friction component which for
convenience may for a first ap-
proximation be assumed constant,
(d) The weight component which exists
when the gun is elevated and is
subtrative since the weight com-
ponent acts opposite to the brak-
ing forces.

(2) With a recoil system where the brake
system is part of or connects with the recuperator
system, we have K * pi+R-Wrsin J0, where p is the pres-
sure in the recoil cylinder. Now, due to the throttling
through the orifice valve between the two cylinders, we
have i a

I C V

P ""Pa a where pa » the recuperator pres-
sure on the oil side of
the recuperator cylinder.

v * the velocity of recoil, (ft/sec)

wx = the opening of the orifice- at the recoil x.
Further, the pressure in the recuperator at recoil
x, in terms of the initial pressure pai«, becomes,

, ( V0 k i 3

o x
Hence substituting in the recoil equation,

394

A)A+R-Wr sin 0

sin

which is of identical form as the equation for re-
sistance to recoil, where the recuperator system is
independent of the braking system.

Thus again, we may consider this recoil system
as having the total resistance to recoil divided
into, the hydraulic throttling, the elastic, the
functional and the weight components.

It is, however, often more convenient to con-
sider the total resistance as divided into "pressure
drops". In considering pressure drops we refer the
pressure intensities to the effective area of the
recoil piston and thus the friction and weight component
drop, becomes,

R-lfr sin

(Ibs per sq.in)

If 8f * the floating piston friction and Aa the
area of the recuperator, the drop across the float-
ing piston becomes, g

pa - pa » -— (Ibs. per sq.in)
*a

Therefore the total resistance to recoil in terms of
pressure drops, becomes

T" p

R-Wr sin 0

c v

, ,
Pa) *<Pa ~ Pa>+Pa

2 Rf R-Wr sin 0

R-wr sin 0

* A

395

STABILITY COH8IP1HATIQM

Now if

Kh • horizontal resistance to recoil

h = height of center of gravity of recoiling parts

above the ground.
wg > weight of the total

ls * horizontal distance from spade to center of

gravity of W3 with recoiling parts in battery.

wc * weight of carriage proper (not including re-
coiling parts)

lc » horizontal distance from spade to center of
gravity of carriage proper.

Vr * weight of recoiling parts.

lr * horizontal distance from spade to center of
gravity of recoiling parts in battery.

e » constant of stability

then since Wslg« WrlP + *clc for any displacement x,
the stabilising moment becomes, Wr(lr- x cos 0)+Wclc=Wslg
- Wrx . Therefore, with a given Margin of stability,
we have, KDh * c(Wsls - Wrx) and hence for a constant
margin of stability throughout the recoil at horizontal

elevation,

e W81, e *r

b b
That is, the resistance tp recoil at horizontal

recoil, should decrease with the recoil consistent with
this equation.

When a constant resistance is maintained through-
out recoil at horizontal elevation, Kh should be limited
consistent with stability in the out of battery position.
Advantage of the total resistance to recoil following
the stability slope:

(1^ More energy is dissipated by the
brake during the powder period, by fol-
lowing the stability slope and thus gives
a greater decrease of the recoil dis-
placement during the powder period.

396

(2) The braking forces being bigber during
the greater part of the recoil, the re-
maining energy or energy of constrained
recoil, is dissipated in a shorter re-
coil displacement.

Hence the total recoil displacement is decreased
over that with a constant resistance to recoil.

Farther, since the stability slope causes a
smaller resistance to recoil in the out of battery
position with a longer recoil, the total resistance
to recoil if Maintained constant throughout recoil Bust
be smaller, and the recoil displacement greater for
a given energy than when the resistance to recoil fol-
lows tba stability slope.

The relation can be shown analytically as follows:
Assuming a constant resistance to recoil maintained
during the powder period and varying with the stability
throughout the remaining part of the recoil, we have
for a variable resistance to recoil throughout recoil,
K0T K0T*

Vr ' Vf ' V; '«• ' B ' T~

•r "r

where KQ = the resistance to recoil maintained constant
during the powder period.
Since c Bgl c

the stability slope becomes, m » - (

therefore, the resistance to recoil in the oat of
battery position becomes, k * KO - m(b-Sr), we have

therefore,

r

Substituting for Vp and Er, we have solving for b,

b, » B * (1- f )± *o (1- f

& in— m 6 T*

where A » - mr ?j » energy of free recoil

397

o

398

c. *r

stability slope

K0

With a constant resistance throughout the recoil,
K(b-Br) - ^ mrV« (1)

KT* KT

where E_ » E - - — and V_ * V* - — -

2mr mr

c W.l_ c W»
and K » — r8— * — E b (2)

a h

Combining (1) and (2) we obtain the length of

recoil for a constant resistance throughout recoil,
and consistent with the out of battery stability,

ca VfT-E i /

bc * T -(— T— >± r- /[•(VfT-EJ-C.j'^BlA+C^B-vyr)]

£t m o o m

where r w l r n

3 S v*TfB

C- * — ; m » stability slope

h h

A * - nrVf energy of free recoil.

bv Length of Recoil for Variable Resistance
The ratio r— * — — — — — — — — — — ——————

bc Length of recoil for constant Resistance

to Recoil
to Recoil

gives the percentage of recoil by following the
stability slope to that of a constant recoil consistent
with stability in the out of battery position.

The relation can be shown graphically, fig.C ).
The ordinates to the line A8 represent the maximum
possible overturning force consistent with stability.
The slope of

399

cWj, C Wglg

AB « — — and the ordinate oA » — — — , Main-
fa h

taining a constant resistance to recoil during the
powder period consistent with stability we have
ordinates to DE, in the powder displacement oH. The
resistance to recoil decreases according to the EF con-
sistent with a constant margin of the stability. The area
OD, EF Go, represents roughly the energy of recoil
A » - mr Vf • If now a constant resistance is to be
maintained we have diagram o J P C where the constant
resistance to recoil o J = C P, and CP = c 8 C, that
is, is consistent with a given margin of stability
in the out of battery position, and further the area
oJPC»A*-jmrVf (the energy of free recoil).

METHODS OP CALCULATING With independent recuperator
THROTTLING ORIFICES. systems, the throttling is

usually either by throttling
grooves or bars or by a mechan-
ically controlled orifice as

in the Krupp valve mechanism previously described.
Let us now consider the necessary throttling
orifice variation along the recoil.

Daring the powder period, we have two methods,

(1) To maintain by a proper variation
of the throttling grooves a constant
resistance to recoil during this
period.

(2) To maintain a constant orifice during
the powder period.

In method (1), we have,

r( ) + Rt- Wr sin £i » K A con-

Vo~AX stant

during

the powder period. Therefore

C A- V

= 13.2 / K - pa - Ft + Wr sin 0

400

where K » — — — for a constant resistance to reooil.
b-E+7fT

K » — ^^— — — — — — — for a variable re-

2[b-B+VfT- 5- — (b-B)J 'Stance to recoil.

tr 2 B-

f Vp .\t

Pa * PaiAv^ ' « paiAv approx. during powder
YQ— AX.

period unless the recoil
is relatively short.

Rt > Rg+ Rp « total friction: guide friction * pack-
ing friction,
further from interior ballistics, av. total powder

pressure w v*
o
Pe a

2g a

w * weight of shell

TO * »u8«le vel. (ft/sec.)

u * total travel up bore (ft)

27 a
Initial pressure on breech in gas expansion po^ = — c

4

1.12 pM (Ibs) where pa » total powder

(b*u)" pres.ure.(lbs)

and

27 P« / 27 P«

e « u(±I D t /i - *L Ji). - !

16 pe 16 pe

3 u wv0+4700 w

t0 " — 7- approx. Vf - —-

* ?o "r

rfo) Wp (W*0.5 w)yQ

*•» ~~^r T Vf°"~r

r

T * t + t « total ponder period (sec)

401

and g = Xf0 + f^ + total free recoil daring powder

period, (ft)

Three points are sufficient for the orifice
curve during the powder period and the corresponding
constrained velocity and displacement to sub-stitute
in the orifice equation with its lay out are:

K lo
7f * Vfo (ft/sec)

"r

lii

2m,

(ft)

when the shot

leaves the muz-
zle.

Ktm
= v -- (ft/sec)

fm

where

'f.*:*

(ft)

> the maximum restrained
recoil velocity and
corresponding orifice.

4.r(frvgo)

T

+ P^bd
•r

K(T-t0)

Pob

6«r(V£-70)

sec.

Rt

- — (ft/sec)

<ft>

At the end of the
powder period.

402

After the powder period, that is during the
retardation period, we have for a constant resistance

to recoil, simply,

and therefore

CA o-

13.2 /K-pa-Rt+Wrsin 0

which gives up the required throttling with a con-
stant resistance to recoil during the retardation

period of recoil.

When the resistance to recoil is variable, we
have during the retardation period, that

1 mrvj -[K - 2 (X+b-2Er)](b-x)

K- £(b+X-2Br)](b-x)

and there- / „,
fore . >^IK-- (b.X-2Br)](b-x)

CA1 / ~T

w , ===^=^==^=:_i.— (sq.in)

13.2 /K-pa-Rt+Wr sin 0

which gives the required throttling with a variable
resistance to recoil during the retardation
period.

With a pneumatic or hydro pneumatic recuperator
system, VQ k

Pa " PaiM» ,A y> where k • 1-1 to 1'3
o v V0- initial

volume.

403

St~ S6
With a spring return recuperator, pa » So » • • • - X

b
b - length of recoil (ft) where So » initial

compression of the
springs (Ibs)
S * final compression of the springs (Ibs) and

••— » 2 approx.
so

In method (2), with a constant orifice during
the powder period, we have

cVv1 dv

* * iTTTf • p« -"**"«• ain * ' "r JT

Since an integration of this equation is complicated
an approximation is made by assuming the recoiling
mass to recoil during the powder period "a" given mul-
tiple distance of the free recoil displacement when the
shot leaves the bore. Let

Er * length of constrained recoil during powder

period, and corresponding length of constant
orifice (inches)

u » total travel of shot up bore (inches)
I * constant from (2 to 2.5) use 2.24
w * weight of powder charge (Ibs)
W s weight of projectile (Ibs)
Pn * total hydraulic pull (Ibs)
wx = area of orifice (sq.in) at recoil displace-
ment x (in)
A » effective arc of hydraulic recoil piston

(sq.in)

c * coefficient of contraction - - - 0.85 to 0.9
d« - S. G. of fluid » 0.849

« a 2 u (1)

Now the total drop in pressure through the recoil
orifice, becomes,

404

7(d0 62.5)A* 7X
p » — — — — — (ibs per sq.ft)(See Hy-

gc wx dro dynaaics)

or

62.5 d0 A» 7»

P * — (Ibs. per sq.in)

64.4 x 144 c'wj

During the retardation period of the recoil, we
have fro* the equation of energy,

K(b-x) » "r . » 64.4 K(b-x)

— — * - 7- hence 7- » ,. '

12 * « 12 »r

therefore, .*,,/u \

62.5 dA K(b-x)

12 x 144 c'wj Wr

and

.
w. -.0361 -^ - (2)

d0A8K(b-Br)
.0361 i*J- - - — (3)

which gives the orifice at any displacement x in terms
of the total resistance to recoil, R and the total
hydraulic pull Ph.

When the resistance to recoil is made to conform
with the stability slope, we have

t 64.4[K-0.5»(b+X-2Er)](b-X)
12 wp

62.5 d0 A*[K-0.5B(b+X-2Er)](b-X)
P

12 x 144 c* w* Wr

405

hence w. * .0361 • — — — —

C* "r Ph
and

d0A*[K-0.5»(b+X-2Br>](b-Er)

.0361

C" Wr

Further

P - K + W .in * - R -

for pneumatic or hydro pneumatic recuperators,
and ss

Pb - 8 + fr sin 0 - 8t - (S0 + — ^ x )

for spring return recuperators.

METHODS OP THROTTLING (1) The simplest net hod

•f throttling is by vary-
ing an orifice through the
piston by throttling bars

fixed to the recoil cylinder

and moving in the apertures through the , piston. Let
wx * the throttling area (sq.ia) as previously

calculated.
S * width of throttling bar or whole in piston

(inobes)
b - depth of hole in piston fro* cylinder surface

(in)

d = depth of throttling bar (inches)
d0 * initial of bar (inches)
n » number of bars (usually n = 2
Then the initial or maximum opening

w0 » nX(b-d0)(sq.in) approx. and for any other
point in the recoil,

wx * nS(h-d) (sq.in) approx,
With grooves in tbe cylinder wall.

wx * n S d (sq.in) where d » depth of rectangular

groove (in)

406

(2) When the throttling takes place around
a long buffer rod of varying diameter

and passing through a circular hole in the
piston, as in the Schneider material,

we have, if

0 « diam. of bole in cylinder (sq.in)
dx» diam. of buffer rod passing through hole in

cylinder (sq.in)
then

»x - 0.7854(D*-dx) which gives the variation of

the diameter of the buffer

rod along the recoil. la the Pilloux recoil mechanism,
we find grooves of varying depth in the buffer rod,
engaging with holes through the piston. The object
of this arrangement is to pass from one set of grooves
to another by turning the buffer rod on elevating the
gun, thus making it possible to shorten the recoil
on the elevating the gun.

If n * number of grooves engaged during the

recoil,
s = width of groove (in) and d = depth of

groove (in)
then wx = n 3 d.

(3) When a constant orifice is main-
tained during the powder period we may
use the so called Krupp valve, which has
bad a wide application for artillery
brakes.

The initial orifice is closed uniformly by a disk
on the piston rotated by a heliooidal groove in the
cylinder wall of constant pitch. Let

00 * initial angle moved by valve disk during

powder period before engaging the throttling
area in the piston.
0t * angle moved by valve during the retardation

period.

p * pitch of helieoidal groove in cylinder wall
(inches) ( Linear displacement per complete

407

408

revolution of disk.)
ro * radial of cylinder (inches)

r * radios to bottom of tbrotting opening con-
tour (inches) then the number of turns »

0

— and the linear displacement x, becomes,

2n

0
x - — p

2x

Hence with a constant pitch with the total recoil
displacement b inches, we have

3xb

hence 2n(b-Br)

0 * '

Further the throttling area becomes, dwx »(— — — )d0
hence * • . r»
wx - / * -2s d0

0

dx (sq.in)

For computation and design it is more convenient
however, to express the throttling area in terms of
the displacement from the end of recoil, since the area
is zero at this point and opens up gradually to its
maximum near the battery position. We have then,

w » /(b"x) Mre~' > d (fc-x) wnepe r , o, where x » b,

In the forn of a summation which lends a simple
practical method of laying out the contour of the
aperture in the piston, we have, if the displacement
of recoil is divided into "n" parts from Er to b,

409

Starting from the out of battery position,
we bare, n^-r^)

*„ * I A(b-X__r) (sq.in)

'n-i

-Xn.) (sq.in)

»a-g M " A(b-X_) (sq.in)

and

*g * ~ 2o(ro~rn-g>A(b~xn-g> Orifice area at

point g from the

out of battery position.

Thus from a step by ste{. process ire lay oat the
contour of the aperture in the piston, and the total
area of the orifiee at any displacement of the
recoil, measured from the out of battery position,
must equal the required throttling area at this
point.

(4) Another form of geometrical

throttling, devised in order to effect
variable recoil consists essentially
of cutting off holes in a perforated sleeve
by the piston, the throttling taking
place through the sleeve in the front and
rear of the piston. We have therefore
two distinctive throttling drops, that
in front of the piston, and to the rear
of the piston through the recoil sleeve.
If wx = the throttling area in front oi the piston at

any point in the recoil (sq.in)
wy » the throttling area to the rear of the pis-
ton at any two points in the recoil (sq.in)

410

px » the drop of pressure through the throttling
areas wx, in the sleeve, (Ibs/sq.in)

Py * the drop of pressure through the throttling

areas wy, in the sleeve (Ibs/sq.in)
We have for the total drop P

P - Px + P,

(assuming the throttling

175 wx 175 Wy constant C the same)
henee

F * — — — — (— — + —

175 «• w;

where *c is the equivalent throttling area and corres-
ponds to the area obtained in the previous throttling
area calculations.
In general

"•I **!**!* »T

when we have a drop of pressure due to throttling through
various orifices in series.

With only two throttling drops, we have

*x "v

we * * ~~-~ and »x+wy * constant.

/ m» + **
wx "y

Prom these two equations, we have at the maximum
value of wa,

"x " *y

Hence in laying out the holes in a sleeve valve,
we place the piston at its displacement corresponding
to naximum throttling, that is at the point of the

411

maximum retarded or constrained Telocity, making the
throttling drop on either side equal.

The process of laying out the required orifices
and corresponding holes is as follows:

(a) At max. throttling displace*

ment corresponding to max. retarded
velocity in the recoil,

Px » P_ « - and wx * wy but since we have a void

in back of the piston due to
the displacement of the piston
rod, P = P i.e. the total drop
3 the pressure in the recoil
..v cylinder.

c A 7

13.2

2

and 2 C A Vxm

13.2

(b ) Next move the piston from

the position of max. velocity,
a unit distance equal to the

width of the piston in the

direction of recoil.
The area to the rear - w_ =

"c

-r , since no openings have been uncovered

m

in the rear.

The area to the front is obtained from the
equations,

~ - — + — where wce, wxo etc. are
ct xi y the throttling areas at
max. velocity and wc^ wx^ etc are the throttling

areas at a distance from

the position of max. velocity equal to the first
unit displacement, hence

412

(sq.in)

(c) Next move the piston another

unit distance in the direction
of recoil, the area to the

w w rear,
c f c

hence wc w

*x * * (sq.in)

(d) Hence for all succeeding points

in the recoil, w-g » wc - wyj-
and y&

"xg ~ —

(e) ID the powder pressure period,

we move the piston backward towards
the battery position from the
position of maximum velocity
succeeding units to the rear and
the process is exactly similar
as moving forward in the direction
of recoil.

THROTTLING THROUGH A With dependent recuperator
SPRING CONTROLLED systems, as in the St. Cbamond
VALVE. recoil system, the drop of pres-

sure between the two cylinders
(i. e. between the recoil brake
and recuperator cylinders* may be obtained by
throttling through a spring controlled orifice between
the two cylinders. A spring valve, however, may be used
with an ordinary recoil brake cylinder, the throttling
taking place through a spring controlled orifice in
the piston.

413

Let p » the pressure in the recoil cylinder (Ibs/sq.in)
a = the area at base of valve (sq.in)
pa » pressure in receiving chamber or recuperator

(Ibs/sq. in )
pai = initial pressure in recuperator (Ibs/sq.in)

Paf = final pressure in recuperator (Ibs/sq.in)
Aa = effective area at top of valve (sq.in)

at * area of valve stem

S0 = initial spring compression (Ibs)

Sf = final spring compression (Ibs)

A * effective area recoil piston

h » lift of valve in inches

c = effective circumference at valve opening
Then, at the maximum opening, giving a lift h, ire
have pa- Pai^a = ^f ^^s^ (approx) and when the
valve is about closed, pa- pafAa = SQ (Ibs ) (approx)
When Aa = A, as with valves in which the spring is
entirely enclosed in the recuperator chamber, we have
(p— pa^)a=S£ when open (approx) and (p-paf)a»S0 when
closed (approx.)

When the spring is outside of the recuperator
chamber, and a valve stem passes through a stuffing
box to the outside of the chamber, we have

pa- PaAa * Pa-Pa<a~ai>=(P~Pa>a+Paai (lbs^

Further at maximum opening of the valve we have
for maximum throttling

!

P-Pai * - T~* "bere C0 = - to -
175 C h 0.6 0.8

hence

C0A V
h = -

which gives the lift of the valve at max. opening
and corresponding to a spring reaction = Sf Ibs.

Therefore knowing p, pai and paf and solving for
the total lift h, we have, for the spring required:

414

Initial load ................ 30 (Ibs)

Final load ................. 3f (Ibs)

Total lift ................. b (in)

Spring constant ............ Sf"So (lbs Per in>

b

which completely specifies the spring required to

properly function the valve during tbe recoil.

Now the pressure in tbe recoil cylinder, is

K+ffr sin t! Rt
p » - (Ibs/sq.in)

A

vo *
and in the recuperator cylinder, p_ » paj ( )

VAX

(Ibs/sq.in)

The maximum throttling opening occurs, at dis-
placement Xm in the recoil, that is at the point
during the powder pressure period, where the powder
reaction just balances the recoil reaction. This
is slightly before tbe end of tbe ponder period and for
an approximation we have,

«

— • where at * 2 approx.

Further tbe maximum constrained velocity may be

taken at, Vr * g 7f where g * 0.88 at short recoil

* 0.92 at long recoil.

Therefore at maximum opening of tbe valve (lift h")
we have,

K+» sin 6 - R Vo

and at tbe end of recoil,
sin ? - R

415

Now due to the hydraulic throttling.

C0 A Vr

/K+Wr sin 0 - Rt V(
1.2 / -

13.2 / pai

Thus we have a complete specification for the design
of the spring. If now, ps = Ks+Wr sin 0 - Rt = pull at
short recoil, max. elev. (Ibs), ph=Kh~Rt = PUH a* long
recoil, zero elev. (Ibs) Fvj = initial recuperator re-
action, required to hold gun at max. elev. in battery
(Ibs), Fvf» » Pyi * final recuperator reaction at the

end of recoil (Ibs)

We have, at short recoil, max. elevation, at the be-
ginning of recoil, p p .

T- a -- I

A A

Sf " T- a -- a

* — ; - a + — — a (Ibs) with
A A *

springs

functioning outside recuperator chamber, and at the

end of racoil, n
P

vi

°

(lbs)

» — a -r— at (Ibs) with springs

functioning out-
side recuperator chamber.

The corresponding max. lift at short recoil becomes,

3

Cl tl E1

OA v Fvi

now Sf-S0 * — aa(«-l)

13.2 c / P3~Fvi
and the spring constant, Ibs. per linear inch, becomes,

Sf-S0 13.2 c Aa(m-l)Fyi / pg-Fvi

S = — 3 - (Ibs. per in)

h i.

C0A» V

416

From the above equations, we see, therefore, that
the load on the spring is large at short recoil and

proportional to the difference of the pull at max.
elevation and the initial recuperator reaction and this
load is increased proportionally to the valve stem
area and load on the air. Therefore to decrease the
load on the springs, the valve stem should be made
as small as possible, only sufficient to carry the
spring load. The lift varies inversely as the square
root of the difference of the pull at max. elevation
and the recuperator reaction, and when this difference
is large as in short recoil, the lift is proportionally
small. Finally the spring constant (that is the slope
of the load - deflection chart) increases with the load
on the air and with the square root of the difference
of the max. pull and the initial air recuperator re-
action. On the other hand, if the compression ratio
is low, approaching I, or if the annular area or the
effective area on top of the valve is small, that
is, using a large valve stem, we must have a spring
of considerable deflection for a given change in
load. When Aa » 0, or Fyi » 0, we have no change in
load in the spring and the valve would open a given
lift h, with a corresponding spring reaction. As the
gun recoils, if the lift and corresponding throttling
area remained constant, the pressure would drop pro-
portionally to the square of the velocity. This, there-
fore, causes a gradual closing of the valve since the

spring reaction must decrease, and we have a
throttling in between an ideal spring controlled orifice
and that with a constant orifice. Even with this
arrangement we have a vast improvement over that of a
constant orifice and the peak in the throttling ia
greatly reduced.

Now, at long recoil, horizontal elevation, at
the beginning of recoil, p p .

Sf = — a - — Aa (Ibs)
A A

417

» — — — a (Ibs) with spring functioning inside

recuperator chanber as is
usually the case at long re-

coil (See St. Cbaaond Chapter), at the end of recoil,

Pb -pv
S0 = — a

A A

bvi
* — — — a (Ibs) with spring functioning in-

A side recuperator chamber.

Tbe corresponding max. lift at long recoil, be-

cones, 3_

C0 a" V

h a ^— — — (inches)

13.2 c / Pn-*vi

Further B .
vi
Sf-S0 = — — a (m-1) (Ibs) and the spring constant,

Ibs. per linear inch, becomes,
S*-SA 13.2 c a(i

C0 a« V

Prom these equations we see the load on the
springs is relatively snail as compared with short
recoil, the deflection b large and the spring con-
stant snail.

Thus, in comparing the requirements of spring
characteristics at short and long recoil respect-
ively, we have,

(1) Short recoil and max. elev. =

A large spring reaction and small de-
flection with a spring constant having
a steep load deflection slope.

(2) Long recoil and horizontal elev.*

A small spring reaction and large deflection
with a spring constant having a snail
load deflection slope.

418

To meet the requirements of (1) in the St. Chanond

recoil system we find Belleville spring used; and in
(2) the use of a weak spiral spring.

When a spring valve is used without a recuperator,
the spring valve is usually located in the piston of
the hydraulic cylinder. In the design and working
of this valve the following points are important:
Let

Phi * the initial hydraulic pull (Ibs)

Phf « the final hydraulic pull (Ibs)

A * the effective area of the recoil piston
(sq.in)

a * the area at the base of valve (sq.in)

Pai = initial recuperator reaction

Paf * final recuperator reaction

Rt * total recoil friction
Then Ph^ + Paj + Rt - Hr sin 0 » K at the beginning

of recoil, and

phf * paf * Rt ~ wr sin ^ * K at the end of recoil,
hence Pni » K + Wr sin 0 - Rt - Pai : Phf»K+Wp sin

At the beginning of recoil,
p

—r- a (Ibs) the pressure in the back of the valve

being negligible.
At the end of recoil,

S0 - a (Ibs)

The throttling at the beginning of recoil, be-

comes

a (Ibs) and the spring

419

13.2 a c /Pbi(Paf-pai>

C0 A * V
The above equations show that the maximum load

on the spring depends upon the maximum hydraulic
load, the assembled load on the minimum hydraulic load
at the end of recoil, the lift varying inversely as
the square root of the maximum hydraulic load and the
spring constant or the compression deflection slope
of the spring being proportional to the difference
between the final air and initial recuperator re-
action and the square root of the maximum hydraulic
reaction.

The spring throttling valve has been used success-
fully with an ordinary hydraulic recoil brake
cylinder, designed for approximately constant pull
throughout the total recoil as in the lower brake
cylinders of a double recoil system or in the brake
cylinders of a gun or sliding carriage mount. Of
course it is impossible to maintain an absolute
constant braking resistance throughout the recoil as
previously discussed but a sufficient approximation
can be obtained to justify its use.

In the design of constant braking with a
spring control, we have a spring valve seated in the
piston.

If the throttling takes place mainly through the
valve seat, we have p a » So * S h where p * pres-
sure in the recoil cylinder, (Ibs/sq.in)

S0 = initial spring load (assembled load)(lbs)

S = spring constant (Ibs/in)

a * the effective area at the base of the valve.
h = lift of valve (inches)

Now coA 7 l l

h * , Ca » to

13.2e/p 0.6 0.8

If the valve is bevelled the throttling area
becomes in place of c h,

w « * D h sin 0

420

whore D * >ean diai. of the bevel portion of the

valve (in)
tf » angle of bevel leasured with respeet to

the central axis of the valve,
beneo CQ A 7 j 1

h - ===-, C = to

13.2 n D sin 0 • p 0.6 0.8

To design the spring we may adjust So to give a suitable
value of the spring constant S, by the formula, -

S >

RECOIL THROTTLING WITH When a buffer or regulator
A "PILLING IN* COUNTER is desired to act through-
RECOIL BUFFER. out the counter recoil, the

counter recoil buffer
chamber must be filled
during the recoil.

The filling of the counter recoil buffer chamber
during the recoil, affects the recoil throttling in
two ways:

(1) The total oil displaced by the
recoil piston does not pass through
the recoil throttling grooves: a
part passing into the buffer chamber
in the process of filling it in the
recoil.

(2.) In the buffer chamber, we have more
or less pressure during a part of the
recoil, since if the throttling into
the buffer chamber is just sufficient
to fill during the max. vol. of recoil,
we will have if the pressure in the
recoil cylinder remains constant an
over filling during the latter part
of recoil and therefore pressure in
the buffer chamber, since the
throttling drop is decreased due to
the decreased velocity of recoil.

421

Therefore tbe total hydraulic reaction
eo the piston rod is somewhat modified.
Let p * intensity of pressure in recoil cylinder

(Ibs/sq.in)

A * effective area in recoil piston (sq.in)
Ab * effective area of buffer (sq.in)
Vx * recoil velocity (ft. sec)
wx = recoil throttling area (sq.in)
ao - entrance throttling area for filling buffer

chamber in tbe recoil (sq.in)
Pb * intensity of pressure in buffer chamber

(Ibs/sq.in)

Then, during the recoil, we have, for the tension in
tbe rod "Ph"

Ph - p A - pb Ab (Ibs)U)
Tbe drop of pressure due to tbrottling through
tbe filling in bole to tbe buffer chamber, becomes

for continuous filling,

_ .1 .« _»
co Ab vx

Ph s P ~ Pb = - (Ibs/sq.in) (2)
175 a»

hence C'*Au V*

Ph = p(A-Ab)* ° X (Ibs) (3)
175 a«

175 a«

(Ibs/sq.in) (4)

A - Ab

Further, ifith continuous filling, we have, for tbe
velocity through the recoil tbrottling orifice,

u-V vx

YX „ (ft/sec)(5)

and therefore, ca(A-Ab)* V*

P ' (Ibs/sq.in)

175 "x (6)

422
Combining U) and (6) we have,

w

a
C0(A-Ab)«

* / C'*A> 7*

13.2 /Pb - n o i
175 a«

which gives the required recoil throttling area
(assuming a density of the liquid = 53 Ibs. cu.ft.) in
terms of the total pull Ph, the recoil constrained
velocity Vx and the constant filling in entrance
area to the buffer chamber ao.

If the density of the liquid is different from
that of hydroline oil * 53 Ibs/cu.ft.we have,

— (sq.in)

ni*ni? v*
288g(Ph -

288 g a«

C0VX /D(A - Ab)'

/ • (sq.in)

12 Cr» DA» 7«

-* *-*-)

288 g a«

where D » weight of liquid per ou. ft.

If we have several contractions in the filling
in passage to the buffer chamber, we have approximate-
ly assuming tne same contraction factor for the flow

°»' 1 1 1 1

— _ = — + — + _ _ _ _ —

» a * x

Determination of ao:

If we desire a continuous filling of the counter

423

recoil buffer chamber daring the recoil, with a constant
entrance throttling area for filling the buffer chamber,
we must design ab for throttling at maximum velocity of
recoil, since the throttling drop varies with the square
of the velocity and is a maximum at maximum velocity, and
the pressure in the recoil cylinder remains approximate*
ly constant during the recoil.

If now, the throttling drop is just equal to the
pressure in the recoil cylinder at maximum velocity, since
the throttling drop is less at all other velocities and
the pressure head the same, we have a pressure in the
buffer chamber continuously rising during the latter
part of the recoil.

If the throttling drop at maximum velocity is less
than the pressure head in the recoil cylinder, we have
a void in the buffer chamber daring the first part of
recoil when the velocity of recoil is large, and there-
fore, not continuous filling.

For continuous filling, therefore pmax > pb at max.
vel. of recoil and therefore max. recoil pressure, that

i« „'*.« ,,«

c» Ab vmax C0 AbVmax

Pmax i a, hence aQ > — —- ,(sq.in)

13.2 / pmax

which gives the proper entrance throttling area re-
quired for filling the buffer continuously during the
recoil.

Since, however, the buffer over fills during the
greater part of the remainder of recoil, ao can be made
smaller than required for a continuous filling through-
out the recoil and yet have a complete filling of the
buffer chamber. In order that the buffer chamber may
completely fill, (though not continuously throughout
the recoil) we have, for the time of recoil, roughly

approx.

PA

424

and assuming the pressure in the buffer chamber at any
tiae of the recoil snail, *

p = — D (Ibs.per sq,ft)
2g

For the filling of the buffer chamber,

= Abb, where b = length of recoil (ft)

a0vt

hence — / ^— > t * Ab b and AQ«CoAbA b /-*— (sq.in)

CQ D *g

where b * length of recoil (in)
Ab = area of buffer (sq.in)

Cn * contraction constant of orifice )= to

0.6 0.8

p = pressure in recoil cylinder (Ibs/sq.in)
A 3 effective area of recoil piston (sq.in)
D * density of liquid (los/cu.ft)

Since the pressure in the buffer is probably small by
this method of filling, we may neglect the total

buffer reaction in modifying the tension or pull in
the rod. Further the throttling in the "filling in"
buffer, becomes, C^AvV8

b¥x

approx

175 a*

hence AbVx * Qb constant - that is, the flow into
the buffer may be assumed constant throughout the
recoil, hence for the main recoil throttling, we

have> C*(AV-Qb)*

175 wj

C0(A7x-Qb)

"x • ~I

13.2 /p

Since, however, pfe (the pressure in the buffer)
actually rises even in this method somewhat towards
the end of recoil, Qj, decreases with AVX and there—

425

fore by slightly modifying the true contraction
constant Co, we have, c AV

wx *

13.2 /~p~

which is sufficiently exact for ordinary design.

For correct filling of the buffer chamber, the
filling throttling area to the buffer should be
made variable. We nay plot this variable area
against recoil and take its mean value as an ap-
proximation for the proper throttling area for
filling the buffer chamber.

The condition for ideal filling of the buffer
chamber are, that C Oxax * AbVx and Pb » 0 throughout
the recoil, where ux = the throttling velocity into

the filling in buffer,

pb * the pressure in the buffer chamber
c = the contraction constant of the

orifice.
ax = the variable buffer filling

throttling area (sq.in)
By Bernoulli's theorem, we have,

DU* D*x

p » — — and p * (Ibs/sq.in)

288g 288g

where vx = the velocity through the recoil throttling

orifice.
D = the weight of the fluid per cu. ft.

A» "

*

and since Pn ~ P A

D Yg

N w — is a variable in the recoil, and therefore

b the recoil throttling areas become modified
at any instant, such that,

426

_ ~v»

bcnee W , _» LJLJL

A*V* c
x *

— A — """^

288 g Pn * C

Constructive difficulties make it impractical
to vary AX according to the above theory in an
ordinary design but by making ax = ao a constant,
and assuming p^ small, we have from the above formula,
that the recoil throttling area equals the throttling
area computed as if no buffer existed in the recoil,
and lessened by a constant area Q

VARIABLE RECOIL:- Stability consideration: As the
VARYING THE RECOIL gun elevates the overturning
AS THE GUN moment decreases, since the per-
ELEVATES. pendioular distance from the

spade point or the point where

the mount tends to overturn on reooil, to the line of
action of the total resistance to recoil decreases on

elevation. Therefore, since the initial recoil energy
is practically constant, it is possible to decrease
the length of recoil considerably as the gun elevates
and yet maintain stability. When the line of action
of the resistance to recoil passes through the spade
point, the overturning moment is independent of the
magnitude of the recoil reaction, and therefore
theoretically the recoil can be made as small as the
strength of the carriage can stand.

427

Therefore, the recoil limitations on elevating
the gun are clearance at maximum elevation, as well
as clearance considerations at intermediate elevations,
and the limitation imposed by stability for various
elevations of the gun.

The recoil may be cut down in any arbitrary
manner provided, that consideration be given to
strength, clearance and stability at all angles of
elevation. The maximum length of short recoil depends
upon clearance considerations at maximum elevation,
while the minimum length of long recoil depends upon
stability at horizontal elevation.

To investigate the stability limitations on the
length of recoil at low angles of elevation, let

C = constant of stability = Overturning moment ,

Stabilizing moment
0.85

Ar= initial recoil constrained energy = - »rVr

(ft/lbs)
Vr=* 0.9 Vj restrained recoil velocity (ft/sec)

w v + w 4700
Vf = = free velocity of recoil

*r (ft/sec)
u * travel up bore (in ft)
Er 'displacement of gun. during powder period 3

(w+ 0.5 w)u
2.25 (in ft)

*r

d = moment arm to line of action of total re-
sistance to recoil (ft)
b = length of recoil (ft)

Then> Ar C0fsls - Wr b cos £1)

b-E,

and solving for b,
we have

/ a dAr
00)- * (Wslg+T*rErcos 0) -4Wrcos(WalsEr*

2 Wr cos $

(ft)

428

which gives us the limiting recoil consistent with
stability for low angles of elevation, with a con-
stant resistance throughout the recoil.

When the resistance to recoil is made to con-
form with the stability slope, we have,

b A

s ~ *r * cos P ) <*X * —

Solving, we have -EWsla(b-EP)

Hence, we have, the quadratic equation in terms of
b

dAp Wrcos 0 t

a ^fr^lg ft * HslsEr - —^ BP

b —

wrcos & lfp cos CT

Solving for b: we have,

Wpcos/)

/ AP

EW81S- /("sis) ~ 2Wrcos 0(— d+WslsEr-

which gives us the limiting recoil consistent with
stability for low angles of elevation, with a variable

resistance throughout the recoil conforming with the
stability slope.

MITHOD Of D1CBKASIHG THE LENGTH OF RICOIL;

In the layout design of varying the recoil on
elevation, it is highly desirable to maintain a con-
stant recoil equal to ihat at horizontal recoil for
the first few degrees of elevation and then begin
cutting down the length of recoil, to the minimum
recoil at max. elevation, since by this method the
margin of stability increases as the gun elevates
and therefore exact stability at horizontal recoil is

429

00

L

430

431

I

/

\

C

(-—• -*

. — -

~ — •

•«

s'

^

$

?

^

CJ

TOM

JO

•01

(D

V)

\L

432

no longer of vital consideration as horizontal fire
in seldom used. In certain types of recoil systems
as in the St. Chamond recoil, the size of the re-
cuperator may be decreased by increasing the pull
at horizontal elevation and therefore in this type
of recoil it is highly desirable to design to the
exact stability at horizontal recoil, as the gun
elevates with constant recoil we therefore will have
ample stability even at low elevations.

Therefore, unless limited by clearance , it is
desirable to maintain a constant recoil from 0° to 20°
elevation, and then cut down proportional to the
elevation to the minimum recoil length at maximum
elevation.

MECHANISM FOR REDUCING Variable recoil is obtained
THE RECOIL ON ELEVATION- by decreasing on elevation

the initial throttling areas
by turning, the counter re-
coil buffer rod which contains

sets of the recoil throttling grooves, as in the Pil-
loux recoil mechanism; or by turning the piston and
its rod with respect to the rotating valve, and thus
changing the initial openings in the Krupp recoil
mechanism; or by rotating a perforated sleeve as in
the American sleeve valve.

Two methods for rotating the throttling rod,
valve or sleeve are used,

(1) by a sliding bar linkage as in the
Pilloux mechanism or

(2) by a four bar linkage as in the
Krupp or sleeve valve recoil mechanism.

With a sliding bar linkage in the elevation of the gun,
a cross head or bar is moved in translation. The bar
contains a pin which engages in a helical groove of
the rotating cylinder, thus giving the necessary
rotatory motion. With a four bar linkage the valve

433

is turned directly in the movement of the linkage
during the elevation of the gun.

(1) In a layout of the sliding bar

linkage, the distance of the translation
of the bar or cross head is fixed by
the pitch of the helix on the rotating
cylinder and the angle turned to be turned
by the cylinder. The pitch of the helix
may not be constant that is the slope
of the helix may vary in the revolution.
With a uniform pitch or slope of the
helix, the decrease in the length of re-
coil against elevation may not be
uniform but for constructive reasons it
may be sufficiently satisfactory.

Knowing the length of the translation of the slide
we may layout the valve mechanism. In the sliding bar

linkage of the recoil mechanism, the crank with
center at the trunnions is made the fixed link, while

the frame of the mechanism rotates on elevation. If

now we draw two circles with centers at the trunnions
and crank pin respectively, the relative displace-
ment of the crosshead or bar is the distance between
the intersection of these circles and a line drawn
through the center line of the slide bar. Constructive-
ly, it is convenient to draw a secondary constructive
circle tangent to the projectile center line of the
initial position of the slide bar, i. e. usually at
horizontal elevation. Then at any elevation the
center line of the slide bar must be tangent to this
circle. Hence the intersection of these tangents with
the base circles of radii at trunmion and crank pin
respectively gives the relative displacement of the
slide. The proper position of the crank pin with res-
pect to coordinates with origin at center of trunnions
can practically only be determined by successive trials
for the proper movement of the slide bar.

434

(2) In a layout of a four bar linkage
the angle of rotation of the valve
during the elevation of the gun is as-
certained from the design of the re-
coil throttling. The gear turning the
valve may mesh with another gear and
from the gear ratio and the maximum
turning of the valve the angle turned
by the valve crank can be determined.
Knowing the angle turned by the valve crank or
valve arm we nay then layout the valve mechanism.
The four bar linkage consists of the frame connecting
the trunnion and valve center; the fixed trunnion
crank connecting the trunnion and connecting rod; the

connecting rod connecting the fixed trunnion crank
and the valve crank or arm; and finally the valve or

arm connecting the connecting rod with the valve
center. Tbe fixed member of the four bar linkage is
the'fixed trunnion crank" joining the trunnion to
the connecting rod. If ire draw two circles from
the fixed centers of the trunnion and trunnion
crank pin respectively, the center of the valve travels
along the circular path with center at the trunnion,
while tbe crank pin of the valve arm moves in a cir-
cular path with center at tbe fixed trunnion crank pin.

It is important to note that the relative position of tbe
valve crank arm should be measured from tbe tangent
to the circle with center at tbe trunnions. Tbe
relative angle turned by the valve crank is therefore
the difference between the final angle with respect to
tbe tangent of tbe trunnion circle when at maximum
elevation and the initial angle with respect to the
tangent of the trunnion circle when at minimum,
usually horizontal elevation.

Constructively, it is convenient to draw a
secondary constructive circle tangent to a
horizontal line through the center of the valve arm.
Then the position of the valve center at any elevation
is the intersection of the tangent to this secondary

435

circle at the given elevation with the base trunnion
circle of the valve.

If He lay off from this intersection the length
of valve arm to the intersection of the trunnion
crank pin base circle, we have the position of the
valve arm for this elevation. For the angle turned
we note the angle made by the valve arm with the
tangent to the trunnion base circle at the valve
center, and the initial angle of the valve am with
the tangent at horizontal elevation. The difference
between these angles is the angle turned by the valve
arm, which multiplied by the gear ratio gives the actual
angle turned by the valve.

ON THE LENGTH OF RECOIL As before for a grooved
WITH A STATIONARY SPRING orifice we have from the
CONTROLLED ORIFICE. equation of energy:

K(b-x)* £ mB v* (1)
where b = length of recoil (ft)

x = recoil displacement (ft)

vx= recoil velocity at displacement x (ft/sec)

mR= mass of recoiling parts

and for the total resistance to recoil, for a dependent
recoil system K = p A + R - Wr sin 0

where p - pressure in the recoil cylinder (Ibs/sq.ft)
R = total friction (Ibs)

A = effective area of recoil piston (sq.ft)

"0* D A*VX D A3?! C V*

P~Pa * - and (P~Pa)A* - =- "
2gc*w« 2gc«w» "x

then since

,, N /0.
Combining (1) and (2), wx =

+ R - »fr sin 0 (2)
2KC(b-x) _

m_(K-paA-R+Wrsin 0)

436

the ratio C * — — — ~ — — — — is approximately con-

K-paA-R+W.sin 0

stant, since the

variation of the weight component Hrsin 0 amd the
recuperator reaction paA is small compared with K.
Then 2CQ

wj =* (b-x) where Co » c'c.

Therefore the orifice variation is a parabolic function
of the recoil displacement and is independent of the
initial velocity and therefore variation in the
ballistics, and is practically independent of the
•eight component and therefore of the elevation of
the gun.

In general, independent of the method of throttling

the length of recoil is practically independent of
variation in the ballistics of the gun or in the
variation of the elevation of the gun.

ON THE LENGTH OP RECOIL During the retardation

WITH A GROOVED ORIFICE, period of the recoil, we

have, from the equation
of energy, -

K(b-x)= ; mr V*

where b = length of recoil (ft)

x * recoil displacement (ft)

Vx= recoil velocity at displacement x (ft/sec)
T.r « mass of the recoiling parts
K = total resistance to recoil (Ibs)

hence

» 2K(D-x).
V » (1)

HOTB: Rot confirmed by observed data. Bditor.

437

D A'V!

(Ibs/sq.ft)

D A*V*

p , p A , » (2)

2gC2W« W«

Ph * total hydraulic pull (Ibs)
A » effective area of recoil piston (sq.ft)
D » weight per cu.ft. of fluid (Ibs/ou.ft)
C = contraction constant of orifice

"herC „ DA»

C »

2gC»

K 2C
Combining (2) with (1), we have W, » — —(b-x) (3)

ph »r

•

If now we assume -— to always remain a constant C1
Ph

a 2C

and placing c C1 - Co, we have Wx » — a(b-x) (4)

mr

which is an equation of remarkable physical significance
We find the orifice variation to be a parabolic
function of the displacement and is quite independent
of the initial recoil velocity. Therefore with the
same weight of recoiling parts, the recoil displace-
ment is practically the same for all values of the
initial recoil velocity. Since the initial velocity
depends upon the ballistics of the gun, we may com-
pletely change the ballistics of the gun and yet with
grooved orifices the length of recoil remains
practically unchanged.

In the following discussion the ratio -r- was as-
sumed to remain constant; the change
in the length of recoil depends therefore on the
change in the ratio H

pb'

Let us examine this ratio for the change under two
conditions,

(I) As the gun elevates where the weight

component is brought into effect.

438

(2) For different ballistics of the
gun, where tbe initial velocity is

changed.
Now for case (1), a V*

K * 0.45 ^— - and assuming the

same length of

recoil, K is a constant and independent of the
elevation.

If K

— is to remain constant, its reciprocal
p
h must remain constant for all elevations.

Since K * Pn+Fy+Rt-irrsin 6 where Ph = total hydraulic

pull (Ibs)

Fv = recuperator reaction (Ibs)
Rt = total friction (Ibs)
Hence

Ph K-Fv-Rt+lfrsin 0 Fv+Rt-Wr sin £1

— * • = 1 - ' '

K K K

Since Fv and K remains a constant for all elevations,
in order that

K ph

or its reciprocal — remain a constant, we must

pb K

have Rt -Wr sin #t= Rt -Wp sin 0g

To consider extreme conditions, let us consider,
horisontal and max. elevation, then

where 0B = the angle of elevation at max. elevation.
Now Rt » R- + Rp where R* = the total guide friction
Rp * the total packing friction

Now Rg is proportional to the total braking * K+lfrsin 0
due to the pinching action of the guides, and the
packing friction remains practically constant since Pj,

does not change greatly. Hence on elevation,

Rt0m > Rt0o usually except for large guns with balanced

palls.

From actual numerical calculations on a series of guns,
tbe term Rto was found to be slightly greater than

439

St0 ~*r sin ^M* Therefore, — remains practically

ph
constant.

(1) The length of recoil with the sane

grooved orifices is practically in-
dependent of the elevation of the gun.

In case (2) with different ballistics, we have
roughly, Kt=* 0.45 mrV*

K * 0.45 mrV*

g

and as before the reciprocal of the ratio — , becomes,

ph ^fy '*rsin 0 h

=- a 1 -- therefore for a constant ratio,

we should have,

pv + fit-Wrsin 0 Py+Rt-lfrsin 0

which obviously is

K K

i * impossible.

But Fv+Rt-Wrsin Of is always small compared with K,
hence the difference of the above terms must be cor-
respondingly smaller.

Hence though the ratio — changes with different

D

ballistics, the change h
is very small.

(2) The length of recoil with the same

grooved orifices is practically independent
of the ballistics of the gun.

ROTI: Tha above disoussion on length of racoil ia
retained as a point for discussion. The
author's conclusion* are not however well
confirmed by observed data. Bditor.

440

COUNTER RECOIL: In the design of a counter recoil
ELEMENTARY system, i»e are concerned with either
DISCUSSION. counter recoil stability when the
gun enters the battery position or
with the buffer pressure in the

counter recoil regulator. In the former, we are con-
cerned with the overall force, that is the total
force towards the end of counter recoil, while in the
latter, with the c'recoil buffer or regulator re-
action. Let

Kv - total resistance to counter recoil (Ibs)
P7 » total recuperator reaction (Ibs)
B^ = counter recoil regulator or buffer force (Ibs)
Rt = total friction (Ibs)

wx = throttling area of c'recoil regulator (sq. in)
C1 = throttling constant
Afc = area of buffer (sq.in)
v = velocity of c'recoil (ft/sec)
The critical angle of elevation for counter recoil
functioning is at horizontal elevation. Then Kv=B£+Rt-Fv
and for horizontal c'recoil stability in a field car-
riage, we have w , + w (b-x)
K

v

h
where lg = distance from total weight of system to

forward overturning point, usua lly the front
\_^L """"" wheel base (ft)

x * displacement in c'recoil from out of battery

position (ft)
b * length of recoil (ft)
h = height of center of gravity of recoiling parts

from ground (ft)

We may express Wslg in terms of the static load on
the spade then, T 1 = HglJ

where 1 = distance between spade and wheel contact with
ground. Then T 1 + Wr(b-x)

where T = 150 to 200 (Ibs)

441

If the ground is assumed to exert a downward
pressure on the spade comparable with the load T,

Ky = 0.85

2T 1 +Wr(b-x)

h

which gives the limitation of the magnitude of the
total unbalanced force towards the end of counter
recoil*

For simplicity in the following discussion a
constant regulator reaction will be assumed acting
throughout the counter recoi I. This method of con-
trol was used by the Rrupp and the earlier material of
the Schneider in France.

SPRIHG RETOHH.

Let S = initial or battery load on spring column (Ibs)
Sf = final or out of battery load on spring column

(Ibs)
Ct = spring constant

Tbenp .8 IT -9

Fvi a so> Fvf ' sf

and the recuperator reaction, in terms of the c 'recoil
displacement x, becomes,

F- » S0 * (b-x)=S0+S(b-x) where S » = the

b b

spring constant,
dv

then mr v -- = -K..
dx

= -(B£ + Bt-Py)
therefore

Brv i Sx

— = - Bxx - Rt x + (S0+S b - — ) x

2 *

which is the general equation of c 'recoil, with a con-
stant regulator reaction and spring return. When
x * b, v = 0, hence

~Bxb~Rtb + (so+Sb"~ 2""^b=0
hence _.

B' = Sft+ ;:— - Rt(lbs)
0 2

This same value may be obtained by a consideration of

442

the potential energy stored in the recuperator.
The potential energy of the recuperator, becomes

b b Sf-S

*o s f S0 dx + / x dx

o o b

sfso b*

V * — T-

b (ft.lbs)

2

We have, then, from the principle of energy,

, VSf
W0 » R^b+Bxb = «— b since Sf = S b + So

ou

hence B, = S_ + — - R*
x o g L

Substituting this value in the energy equation *

2 r

and siaplifying, we have nrv = Sx(b-x) hence S = '

i mrv* b (b-x)x

..d Bx -(80-Rt)* J^^-

which gives the value of the constant regular reaction.

Bx * - Clbs) where C = the reciprocal

of the contraction
factor of the regulator orifice.

Ab * effective area of buffer

wx = variable regulator orifice, and since,

» S(b-x)x
v a

»r

C*Aw s(b-x)x

Bx

175.rBx

and therefore w2 = _____ (fcx-x* )

•»;

Value of

of regulator (sq.in)

443

C Ag /-s
where CQ = *

13.2 / mrB'
r x

BX - V ^ - Rt

The unbalanced force of c'recoil, becomes,

dv ,_«
mr v - • (Bx + Kt - Fv;

dx

= - (S0 + — - S0 - Sb + Sx)
2

= — - Sx = S( - - x)

2 2

Hence the unbalanced force decreases with the dis-
placement of c'recoil, reverses to a negative value
at mid stroke.

The initial unbalanced force at the beginning

of c'recoil, equals

Sb ,sf"so sf"so

*2 = (~!b~ ~
The overturning force at the end of c'recoil, becomes

Sb _ sf~so
.2 2

GENERAL EQUATIONS The functioning of counter recoil
OP COUNTER RECOIL, may best be studied by a consideration
of the physical aspects of the
dynamic equation for counter re-
coil. Let

pa = intensity of pressure of the oil in the air

cylinder (Ibs/sq.in)
"ax a counter recoil throttling area between air

and recuperator cylinders (sq.in)
Ay = effective area of recuperator piston (sq.in)
KV = total resistance to counter recoil (Ibs)
Fv = actual or equivalent recuperator reaction
at any displacement "x" from the out of

battery position (Ibs)
wx = variable buffer orifice at c'recoil dis-

444

placement x for buffer counter recoil

throttling (sq.in)

Then during the counter recoil for a spring, pneumatic
or similar recuperator system, we have,

(1) the recuperator reaction acting to
displace the gun forward into battery
Fv (Ibs)

(2) the weight component resisting Fy - -
Wrsin 0 (Ibs)

(3) tlie guide friction Rg = n Wr cos 0
approx. since the pinching action of the
guides is small on counter recoil and

we therefore have an approxination of
pure sliding friction throughout the greater
part of counter recoil. This reaction
also resists Fv. ,

(4) the packing friction Ks+p resisting
Fv (Ibs)

(5) fhe throttling through fhe return of
the recoil apertures together with the
counter recoil buffer throttling. The
throttling through the recoil is small
as compared with the buffer throttling
and may be neglected or else included
with the buffer throttling. The
throttling is proportional to the
square of the velocity of counter re-
coil and inversely as the square of the
throttling orifice, that is, the buffer
braking becomes,

I »
I COV

H. = - (Ibs) and resists Fw
* a «^________

%
A

Therefore, w« have

n

Fv-Wr(sin 0 * n cos 0)-R8+p " ""

which is the differential equation of counter recoil.

445

With a hydro pneumatic recuperator system it is
possible to regulate counter recoil by lowering the
pressure in the recuperator cylinder for the greater
part or the entire recoil, by throttling the oil
through an orifice between the air and recuperator
cylinders. Introducing a buffer chamber in the air
cylinder with a buffer attached to a floating piston,
gives a simple means for varying the orifice and thus
reducing the pressure in the recuperator cylinder or in
the recoil cylinder to a value consistent for the
proper movement of the recoiling parts in counter re-
coil .

The pressure in the recuperator cylinder due to
throttling through the orifice between the air and

recuperator cylinders, becomes,

iii a
, co v
Pv s Pa - W2
wax

Hence, for the motion of ths recoiling parts in
counter recoil, we have,

' 2

PvAy - Wr(sin 0 + n cos 0) - Rs+p - "g = mp v -— •

wx dx

or substituting for pv, we have

Cn i

. C 2 dv

Pa-Ay - Wr(sin J0+n cos &)- Rs+p~ (— •*• * ~~ ^)v = airv — (2

w?., w? dx

where Co = A^o' '

i

Now p AV may be regarded as the equivalent recuperator
reaction, that is Fv = pa Ay and further assuming
the regulation to be entirely effected through the

throttling in the recuperator, we have, for eq.(2)

n

Crt A ,r

Fv -Wr(sin0+n cos 0)-Rs+_ - — v?=mrv — (3)

"ax dx

which is exactly similar to the previous equation of
counter recoil for a simple spring recuperator system.
The external force on the total mount, is

446

i dv

Kv » »r v — , together with the weight of the

recoiling parts Wp.
During the acceleration,

Kv = mr v -— acts towards

the breech, and
during the subsequent retardation,

KV * Br v d7 aots

towards

the nuzzle. During the acceleration Ky is necessarily
always less than K the total resistance to recoil since,

~Cv*
K = F..+R + -T— - W_ sin 0, for the recoil and

w* — — — — — _— .

cV

Ky = Fv -R - — j— - Wr sin 0, for the counter recoil,
wx ' '

therefore 2 i 2

Cv C v
K-KV = 2R + -j- + — — , roughly assuming total

friction the sane on

recoil and counter recoil. Hence, in the design of a
counter recoil system we are only concerned with
counter recoil stability, and not at all with the re-

action during the acceleration. If we let, further,
Ws = weight of total system (Ibs)

ls - horizontal distance fron front hinge or con-

tact of wheel and ground to the center of
gravity of the total system in battery (ft)
C = constant of counter recoil stability

Overturning counter recoil moment.

Stabilizing counter recoil moment.
i
d = perpendicular distance from front hinge or

contact of wheel and ground to line of
action of Ky through center of gravity of
recoiling parts (ft)

then, for stability at angle of elevation 6, we have

0

s + Dr cos0)-Fv =

(2)

447

dv w-3l3+Wr(b-x)cos 0

and - •- v — • = C t 3 (3)

dx d1

which gives us the velocity curve against displace-
ment consistent with counter recoil stability. Sub-
stituting v in (2) enables us to determine the variable
orifice wx consistent with counter recoil stability,
since Fy is a known function of x.

During the acceleration, we have

°;>* t,

Pv-Wr(sin 0 + n cos 0) - Rs+p ~ " s Br v —

and since we are not concerned with stability, for

• inisuiB time during the acceleration Ky should be made

a maximum, that is the hydraulic braking tern
should be made zero, hence

cV

dv
Fv-Wr(sin 0+ n cos 0) -Rs+p = »r v —-

Let further vm = aaximun velocity of counter recoil

(ft/sec)
xm = corresponding displacement to

maximum velocity from out of battery

position (ft)

Then, for ideal counter recoil, that is the counter
recoil functioning in nininun time and consistent with
stability, we have,

o i b

~ / «r v dv = — / [Wslg*Wr(b-x) cos 0)dx (5)

from which we obtain,
a •

Brvm C

« * «
To determine xn, we have

448

/•x» vm

'Fv " wr(sin Of + n cos 0)- Rs+p]dx = / nir v dv

hence

X

. IE P

Fv dx - [Wr(sin0 + n cos 0)+H8+pl«

— — - cos 0 ] (6)

2

and knowing Fv as a function of x, we may solve for
xm. Substituting in (5') we easily obtain vm which
gives the maximum velocity of counter recoil.

Thus we see during the acceleration it is de-
sirable to make, Kv a maximum, that is

Kv = Fv-Wr(sin 0 + n cos 0)- Ro+n
vmax

and during the retardation Ky should be consistent
with counter recoil stability, that is

dv

cos

which can be obtained by increasing the buffer or
counter recoil regulator, such that,

G'V* W gl'+W (b-x) cos 0

0+W cos 0)-F = C[ - - - ]

A simple graphical solution of the above analysis may
be made as follows:

Lay off the recuperator reaction Fvf-Fv^ and
from the ordinates of this curve subtract Wr(si"n J0 •»•
W cos 0)+Rs+_ which gives the unbalanced reaction
proportional to the ordinates to AB, during the ac-
celeration period. Draw in below 00', CD parallel to
the counter recoil stability slope Q R, such that

— = — = C , the constant of counter recoil stability

assumed. Then we locate M such that
the area OABM = area M Of D C. Since OABM is pro-

449

w/m

+VJC0S t) y- >PJ + p

C 'XECO/L ENEffGY PL OT5

COS

MO 'DC

450

oortional to the work done during the acceleration,

we have

Area 0 A P M = - Mp Vm

•
The velocity curve may be constructed graphically

since any increment area abed is proportional to the
change of kinetic energy, that is

a. i* a
Area a b c d I - mr(vt-vj)

and thus knowing the previous velocity, we may con-
struct a velocity curve directly.

The energy equation of counter recoil:

The dynamic equation of counter recoil, is

cV

Fv-(n cos 18 + sin 0)Nr-Rs+p — * mr v -—

wx dx

where Fv = the recuperator reaction
Rg+ = total packing friction.

&

= hydraulic buffer resistance

"*

x CQv v

Integrating, we have / (Fv-»rsin0-Rt- — y— )d**/ mrv dv

o wx o

where Rt = a W_ cos 0 + ^s + n , ,

x x C0v y2

Separating, we have / Fydx-(Wrsin0+Rt )X-/ — - — dx=mr r-
o o wx 2

FJow since the relative energy in the recuperator, de-
pends only on the position in the recoil, we have,

dW
Fw » - — since ¥v dx = - dW

de

where W is the relative potential energy of the re-
cuperator, which is equal to the work of compression
(approximately) for a displacement in the recoil (b-x)(Fy)

If W = the potential energy of the recuperator in the
out of battery position,

451

• i i a

r * dW Cov mrv

- / — . dx - (ff-sin 0+Rt)x - -2— dx = -£—

W dx w 2

from which we obtain

_

CQV

(*t- Wx)-(«rrsin 2T+Et)x - / - dx )= -

"x 2

which is the general energy equation of counter recoil

Obviously at any displacement in the counter recoil x,

- ' « *

CQv rarv

If, +(Wrsin 0+Rt)x+ / - dx + - = W. a constant

** ** 4 f\ O MH^HWMMM^^HMM^^^K

»x 2

That is, the total energy at any recoil x, is divided
into the potential energy of the recuperator, the work
done by friction, the work done by buffer throttling
and in the kinetic energy of the recoiling mass.

Between any two displacements in the counter re-
coil K^ and xa we have, approximately, provided the
points are sufficiently close:

which gives us a method of computing vx knowing vx
from the previous point.

COMPUTATION OF With a given set of counter
COUNTEE RECOIL. recoil orifices, the velocity
and force curve of counter re-
coil may be calculated by either of
the two following methods:

If Fv = actual or equivalent recuperator reaction at
any dis placement "x" from the out of battery
position (Ibs)

F?i = initial recuperator reaction (Ibs)
wx = variable orifice for counter recoil throttling
at displacement "x" from the out of battery
position (sq.in)
CQ = counter recoil throttling constant

452

n = coefficient of guide friction
Rs+p = total c'recoil packing friction (Ibs)
Ay = effective area of recuperator piston (sq.in)
VQ = initial volume of recuperator (cu.in)
x * counter recoil displacement (ft)

METHOD I - LOGARITHMIC METHOD.

The dynamic equation of c'recoil, becomes

cV

Fv - Wr(sin 0 * n cos 0)-Rs+ — = mr v —

"x d*

If we let, R = ffr(sin 0+ n cos 0)+Hs+p

GovZ dv

then F - R = m_v — -

wx dx

Now Fy and wx are both functions of x and therefore
the equation cannot be readily integrated. If,
however, we take a small interval Fv and wx may both be
assumed constant during this interval. Considering
any two points x^ and xa in the counter recoil,
we have

*a v2 nrv dv

dx = / where A = Ftf-R

Rearranging, we have

C0v»

d(A- -2—
x * w* i

2°i o!,'

A—

hence integrating, we find

453

and

* loge(A -

m

r "x.

therefore

. . o2t

log (A -- )' log (A -- ) --

wx' "x1 2.3nrw«i

2 a *s

where A = Fv-Wr(sin 0+ n cos 2J)-Rs+p (Ibs) fro«
which ire nay construct the velocity curve.

The advantage of this method is that a small
variation of Fv and v»x has a negligible effect on the
equation of motion and therefore fairly intervals nay
betaken provided the throttling orifice of counter re-
coil is not changing rapidly. During the buffer period
where the throttling changes rapidly small intervals

oust be taken.

The total unbalanced force acting on the recoiling

parts during counter recoil, is

dv Av ,
mr v — = mr v — (approx.)

From this the unbalanced force (total accelerat-
ing or retarding force)

Fv-Wr(sin 0 + n cos (? )-Rg+p

x
•ay be calculated and plotted.

To compute the recuperator reaction at any
point, we have for spring recuperators,

srso

Fv = S0 + -^-£-(b-x)
b

454

where SQ * initial or battery spring reaction (Ibs)
Sf = final or out of battery spring reaction

(Ibs)
and for, pneumatic or hydro pneumatic,

Vo k
FV ' Pai V V0-12AV (b-x)

V k
V0

where b= length of recoil (ft)

x = c 'recoil displacement from out of battery
position (ft)

VQ= initial volume (cu.in)

To compute Rs+p, we have, RS+D=100 to 1-50 Zd for
ordinary packing

where d = iiam. of any one of ths various recoil rods
(in)
Rs+p=Z(Ct+Cap)»Z[0.15(.05 *WpdpPlBa3t)+0.75(.05TiWpdpp)]

(Ibs)
where w = width of the various packings (in)

dp = diao. of the annular contacts of the various
packings (in)

pmax = the design pressure, usually the max.
pressure in the cylinder to which the
packing is subjected to (Ibs/sq.in)

p = actual pressure during the various points
in the counter recoil to which certain
parts of the packing are subjected to
(Ibs/sq.in)

Obviously since p is variable, Rs+p must be variable
daring the counter recoil but aq average value of p
•ay be assumed and the corresponding Hg4p can be used
with sufficient accuracy. t

I V

Computation of the throttling resistance C0—

455

(1) with a filling in buffer, the

counter recoil regulation being effective
throughout the counter recoil:

we may neglect the small throttling through the

apertures of the recoil orifice, and then,

* Cf2A!v*
i v Abv

co T (Ibs)

175w»

where C' = the reciprocal of the throttling constant
Ab * area of the buffer (sq.in)
wx - buffer throttling area (sq.in)

(2) with some form of spear buffer,

the buffer action being effective only
during the latter part of counter re-
coil,
we have three stages:

(a) the void displacement with
no regulation.

(b) throttling through the recoil
apertures which cannot be
neglected due to the much higher
velocity of c 'recoil than in
case (1).

(c) throttling through the buffer
orifice, the throttling resistance
being large as compared with the
resistance due to throttling through
the recoil orifice, the latter
being neglected.

In (b), we have,

* f . »3 *

(A+ar) v

co _7

175 w$

xr

where A = effective arc of recoil piston (sq.in)

ar = area of recoil rod (sq.in)

wxr~ area of recoil throttling grooves (sq.in)

In (c), we have, as in (1)

456

• «* »' *

, / B C V

° w« 175 w»

where Ab = area of buffer (sq.in)

wx = buffer throttling area (sq.in)
With a hydro pneumatic recoil systen,
In this type it is possible to loner the pressure in
the recuperator by throttling through a constant
orifice.

Now it has been shown, that

At the end of recoil if a spear buffer in the recoil
brake cylinder also functions,

o

y = the effective area of the recuperator
piston (sq.in)

w0 a the c1 recoil throttling area between the
air and recuperator cylinders (sq.in)

METHOD II - THB HHBRGY KUTHOD.

From the energy equation, we have, for any
arbitrary interval,

» a

Vx2~vx
(Wx»-Wx2)-(Wrsin 0+Rp)(w2-xi; -- - (x,-xt )»mr( - ;

i a

Cov

where Wxn - the recuperator potential energy at the

point "n" in the counter recoil (ft.lbs)
To compute Wxn we proceed as follows,
With a spring recuperator,

•

o — O

"xn = -^[so * ^ — —(b-x)J d(b-x) (ft.lbs)
b

457

= S0(b-x)+ 5f ° (b-x)' (ft.lbs)

2b

where So = initial spring recuperator reaction (Ibs)
Sf = final spring recuperator reaction (Ibs)
b = length of recoil (ft)
x = displacement in counter recoil (ft)

With a pneumatic or hydro pneumatic recuperator,

b-x b-x V0 k

**n = * Fvd(b-x)=F¥i / ( - ) d(b-x) (ft.lbs)
o o V0-AY(b-x)

where k = l.Koil in contact with air)

= 1.3 oil and air separated by floating piston

or pure pneumatic)

AO = effective area of recuperator (sq.ft)
VQ = initial volume (cu.ft.)
Pvi= initial recuperator reaction (Ibs)
Integrating, we have

xn

Av(k-l) y"-1 V*-1
where V = Vo-Av(b-x). Further since,

Pai Vj = PaVk or ^.. A"
Pai V

then, pai , , k

k 1

P V - PiV

aio

Hence, when V is in cu. ft., Av in sq. ft. and b-x in
ft, we have

V = V0-Av(b-x) (cu.ft)

Vo k
Fv = F>vi^ (lbs)

Wx= / ;; (ft.lbs)
Av(k-l)

458

Usually it is more convenient to express V is in cu. in.,
Ay in sq. in and b-x in ft.

V = V0-12A?(b-x) (cu.in)

*V Fyi(— ) (Ibs)

V

12Av(k-l)

(ft.lbs)

To compute Fy, we have log = k log — , a linear

Fvi v logarithmic

equation and therefore may be readily plotted. There-
fore, vie may make a table for computation of the
potential energy of the recuperator as follows:

V wx

X

12Av(b-x)

V

Vo

F
rv

F V-F • V

r VT rVl VO

K lOg ^

12Av(k-l)

459

We have, from the energy equation
W0-Wxi-(Wrsin0+Rp)xt •

+m_ —

s »,

O 2 2 3

Hrt-W_s-W_sinjft+RD)(x -x )- — — + mr — = m. —

XX r 3 * _2 . *0 O

n-1

'xn

The solution of these equations, may be put in a table
form:

X

»,

WX,-V»X2

(Wrsin0+Rp)Ax

"x

^y

2

Vn-l

X

n

v

o

»0

«,

Wxi

«0-«».

(Wrsin<?+Rp)xt

«x'

m^

v,

«,

^

l^!..

(Wrsini?+Rp)
(x,-xt)

X2

C v

_ o t

2

"x'

2

2

V

"•""I

m i

',

«,

v

V-v

(xa-x2)

wx2

c'v;

wSz

X3

•

2
V

r2~

2

m ^

:

xn

o

WX(n-l)

(Wrsin0+Rp)

"xn

C0v2n-l

"xn

o

o

From the above table we may plot the velocity curve,
To obtain the unbalanced force (accelerating or re-

460

tardation force of c'recoil)we have,

cfv* v* _va
Fv-Wr(sin 0+n cos 0)~Rp = mr(— -) (Ibs)

RELATIVE ADVANTAGES OF THE LOGARITHMIC AND ENERGY METHOD
FOR COMPUTATION OF COUNTER RECOIL:

In the design and computation of a c'recoil system,
we are either concerned with counter recoil stability
which is the primary limitation on c'recoil for small
caliber mobile carriages, or with the maintaining of
a low and constant buffer pressure, where c'recoil is
no longer a consideration and the potential energy of
the recuperator is large, as in large caliber artillery.

In the former case, it is import-ant that the
total unbalanced resistance to c'recoil or the total
retardation towards the end of counter recoil, either
remain constant or follow the c'recoil stability slope.
In the latter case, however, it is important to maintain
as low buffer pressure as possible and thus a constant
buffer resistance is used in spite of the total resist-
ance to c'recoil rising towards the end of c'recoil.
In the calculation of the total accelerating or retard-
ing force in c'recoil, the logarithmic method and the
simple dynamic equation of c'recoil are preferable since
we are only concerned with the total unbalanced force
on the recoiling mass. During the first period of c'recoil
a constant throttling orifice is usually used for reg-
ulation and large intervals may be taken by the
logarithmic method. During the retardation the total
resistance to c'recoil is usually constant and there-
fore we have the simple dynamic relation of a mass
being brought to rest by a constant force. With a con-
stant buffer force, the energy method is preferable
since the work done by the buffer and corresponding
kinetic energy and therefore the velocity of c'recoil

461

can be quickly estimated.

Estimation of the buffer resistance of c'recoil,
with constant buffer force and corresponding
velocity of c'recoil:

(1) If the buffer force acts only during
the latter part of c'recoil, we have,
three periods:

(a) the accelerating period, cor-
responding to the void displace-
ment.

..... • . ; . r- .:*<c*/w 8OJ TO:

9
*a

(W0-Wa)-(Wrsin 0+Rp)Xa= mr — - (ft.lbs)

(b ) the retardation period where
throttling takes place in a re-
verse direction through the re-
coil apertures only.

I 2

(W -wb) - (WrsinlZf+RD)(Xb-Xa)- /c! -^- dx = — (vj - va)

H " v w ^ 9

*a "x

xb ^ov
If we neglect the term, / — ~ dx as small, we

xa have immediately

2

(ffo~ffb)~(Hrsin 0+Rp)Xb=mr —» (ft.lbs)

(c ) the retardation period where
the running forward brake or
c'recoil buffer comes into action:
assuming a constant buffer force,
we have „ i 2

~fr = Bx

and

2

r —
2

462

*

vb
Substituting for (-mr — •) from the previous equation,

we have

Bx(b-xb)*W0' Ofrsin 0r+Rp)b

be nee

Wp-(W sin 0*R_)b

BX» — E — <lbs)

b-*b

If Ab« the area of the buffer (sq.in)
db» b-xb » length of the buffer (ft)

b » length of recoil (ft)
pb* the average buffer pressure (Ibs/sq.in)
then we have for the average buffer pressure,

W0-(Wrsin

p *

where

wo * ' pvf * " Fvi

Vf3Vo-Avb m » ratio of compression

To compute tbe velocity curve during tbe buffer action,
we have

x G'V* *r • «

tfb»Wx-(Wpsin Or+Rp)(x-xb)- / -^-dx - ^~(vx-vb)

xb "x

Since Wx and v vary at each point, tbe above equation
•ay be divided into a step by step process, i. e.

Wb-Wx,-(Wr«in 0+Rp)(xt-xb)- j~- (xt-xb)= -| (»Ji-vJ)

^I(x -x )- i v',-v'

463

Pron the velocity curve and buffer pressure pb

(2) Where the buffer force acts

throughout the c'recoil.

At the beginning of c'recoil, the recoil apertures
are snail and the throttling through them during the
c'recoil cannot be neglected. Since, however, this
additional throttling is effective only for a short
distance at the very beginning of counter recoil we
have as a close approximation, for the average
buffer force of c'recoil (assumed constant)

WQ-(WTsin

where as before, WQ= A (k,1) : %f •%!«

COUNTER RECOIL SYSTEMS. Counter recoil systems may

be broadly classified into:
(1) Those in which the brake
comes into action during the
latter part of counter recoil.

(2) Those in which the brake is effective throughout
the counter recoil.

With (1) we have, usually some form of spear
buffer which comes into action towards the end of re-
coil.

With (2) we have, usually a "filling in" type of
buffer, the buffer being filled during the recoil and
acting throughout the counter recoil.

Type (2) gives obviously far better counter re-
coil regulation than with type (1) where in the latter,
we have considerable free counter recoil and corres-
ponding high velocities before the buffer action takes
place. This is especially true for long recoil guns.

(1) Counter recoil systems, where the brake

is only effective during the latter part
of counter recoil. The counter recoil

464

functioning may be divided into three
periods:

(a) The acceleration period during
the void displacement.

(b) The retardation period where
throttling takes place in a reverse
direction through the recoil
apertures only.

(c) The retardation period where
the running forward brake comes
into action.

During period (a), we have

Fv-Rs+p-Wr (sin 0 + n cos 0) = rar v — - and the void
displacement . v

x = !£ (ft)

03 A

where aP = area of recoil rod (sq.in)

A = effective area of recoil piston (sq.in)
b = length of recoil (ft)

hence

f oa mrva

-- - -

0 [Fv-Rs+p-Wr(sin0 + n cos
As an approximation, we have

Fvf+Fva *a xa mrva

( - ) - [Rs+_+Wr(sin 0 + n cos 0)] = -

2 2

where Fvf = the max. recuperator reaction

pva = the recuperator reaction at the end of
the void displacement.

(Fvf+Fva)-2(Rg+ +Wr(sin0+n cos0)]Xa

f -
mr

(ft/sec)

where va is usually the max. velocity of counter
recoil. During period (b), we have

t

Fv ~ Rs+p~ffr(sin 9 + n cos 0) -- - — = m v —

w^ dx

A
*

465

V

where — — = hydraulic braking reaction due to
throttling through the recoil

iiii ' "fii *» r,n ' ' -

apertures.

Now the constant CQ is different from that of
recoil since the area of displaced fluid and con-
traction of orifice on the return motion are dif-
ferent from these factors in the recoil. However,
for a first approximation, we may assume CQ the same both
in recoil and counter recoil. If Va is the velocity
of recoil, with total hydraulic pull P^ at displace-
ment b - XQa in the recoil, we have

2 2

C0v va

s "^7 Pb approximately

and therefore, approximately,

Fva+Fvb va

(— ) - Ks+p-Hr(sin 0+ n cos 0) - — Ph(xb-xa) =

where Fva = recuperator reaction at end of the void
Fyjj = recuperator reaction at entrance to

buffer.

Prom the above equation, knowing va we may readily
compute V{j» During period (c), we have

2 _ I 2

CQv CQv dv

-V?r(sin0+n cos 0) - — - — - — — 3 mr v —
wh "x dx

where — — = the hydraulic braking due to the

W

counter recoil buffer.

Now the term n * n ' 2
o °o
' • is small compared with

"6 u' . *x

and may be neglected, especially during the latter
part of period (c).

Period (c) is the critical period of the
counter recoil since the reaction,

466

-fl — * Wr(sin 0+n cos 0)+R3+p-Fv = — —
t

where = the max. stabilizing force in battery.

d1

The buffer throttling should be designed, either with

cov* , "sis

-Kv = — r~ * Wr(sin 0 + n cos 0VRg+ -Fv= Cs — -

d1
a constant or

r '»* w 1 '
uo i SAS
- Ku = — — + H_(sin0«-n cos0)+R.4.n-F_=C(, ( — )+Wr (b-x)cos 0

V * * 2»Tp » o I

"x

that is consistent with the stability slope of counter
recoil. By the latter method the buffer action nay
have a somewhat shorter displacement in the recoil and
yet maintain the same factor of stability as in the
former.

At this point it is well to emphasize, that a
constant buffer resistance is entirely inconsistent

with counter recoil stability since the total counter

recoil resistance becomes greater in the battery
position than at the entrance to the buffer. Therefore,

a longer buffer is required, for the same mean re-
sistance to counter recoil.

(2) Counter recoil systems, where the

brake is effective throughout the recoil.

In this type of counter recoil,

it is customary to regulate the maximum velocity at-
tained during the acceleration period to a low value,
by the use of a constant orifice throughout the ac-
celerating period. A constant orifice during the first
period of c 'recoil has distinctive advantage since it
gives a satisfactory control together with simplicity
from a fabrication point of view.

During the latter part of the counter recoil, it
is obviously necessary to introduce a variable orifice
in order that the recoiling mass may be brought to rest

467

gradually. We have, therefore,

(a) The accelerating period with
a constant orifice.

(b ) The retardation period with a
variable orifice.

We have the two systems of regulation:

(1) By a buffer brake control in the
recoil hydraulic brake cylinder
throughout the counter recoil.

(2) By lowering the pressure in the re-
cuperator cylinder by throttling through
an orifice between the air and re—
cuperator cylinders.

With simple spring or pneumatic recuperator systems
we must use a regulation system similar to type (1).
With hydro-pneumatic recoil systems we may use type (2)
alone, as in the St.Chamond or Puteaux brakes, or a
combination of type (1) and type (2) regulation, as
in the Filloux and Vickers recoil mechanisms.

In either type (1) or type (2) regulation, for
the running forward brake effective throughout counter
recoil, we have, exactly the same characteristic
dynamic equation.

With a simple recuperator of a s-pring or pure
pneumatic type, we have for the equation of motion,

CV dv

Fv~Wr(sin 0 + u cos 0) - Rs+p — = mr v -—

"x ax

whereas with a hydropneumatic, assuming the pressure

lowered in the recuperator cylinder by throttling
between the air and recuperator cylinders, we have,

_ I I I 2

i co v dv

<Pa ~ ~ )Av-Wr(sin 0+u cos Of) -Rs+p = mr v —

and if we let pa AV=FV« the equivalent recuperator
reaction

468

_ i i i * - " a

C0 * cov

— g "• - • • Ay = — — we have, as before

wtr wv

-." «

cov dv

Fv-Wr(sin 0+u cos 0)-Rs+p - •• = mr v —

w* dx

Further, for the critical condition of counter re-
coil stability, that is counter recoil at horizontal
elevation, in type (1), the reaction on the carriage,
consists of:

(1) Fy acting to the rear

(2) in Wr + Ro + n acting forward

— ~ acting forward

It may be easily shown that the resultant of
these reactions aots in a line, through the center
of gravity of the recoiling parts, the effect of the
reaction of the guides being to transfer the various
resistances and pulls to the center of gravity of the
recoiling parts.

In recoil systems of type (2), the reaction on
the carriage consists of:

iii a

(1) (pa ; )Ay acting to the rear

"v

(2) n Wr + RS+P acting forward.

The line of action of the resultant as before
passing through the center of gravity of the recoil-
ing parts. But the effect of

ilia -,"a

(pa - a )AV is exactly the same as Fv - ~ —

wv "v

cV

^«^
where Fv » PaAv and

„

C0V

" y ''y

Hence so far as the reactions on the mount and motion

469

are concerned, the hydro pneumatic and spring re-
turn types of recuperators are exactly similar.

For the first period of counter recoil: As-
suming a constant orifice during the first part of

recoil, we have

„ ' «

C v dv

Fv-Wr(sin0 + u cos 0) - Rs+p - ' = mr v —

"o dx

If we let, R = (Wrsin 0+u cos)+Rs+p then

C v dv
Fy-R - — = mr v — Now Fy is a function of

WQ dx x, and therefore the

equation cannot be

readily integrated. But since Fv does not vary
greatly over a short interval, we may assume mean
values of FV for a few given intervals. The ad-
vantage of the integration of the equation, is that
we may greatly reduce the number of intervals as com-
pared with that of a step by step process and obtain
sufficiently exact results.

Considering any two points x± and xa in the
counter recoil, we have

x v m_ v dv
/ dx = / -— 2 where A = Fy-R

X V « ^Ov

xt vt A »--•

wo

Rearranging, we find cv

d(A~ ;T"'

x mw *o

hence

2 2 a

mug f* tr /* it

v* ™ *N W V W T

and

470

* * *

mrwo wo o

therefore

C0v*, C0v* 2C(x,-xt)

iog(A - -2-1)

2.3mrwQ

where A = Fy-Wr(sin 0+u eos 0)-R3<.p (Ibs)

Prom this equation knowing the velocity at the
beginning of any arbitrary interval and with the mean
recuperator reaction for the interval ire may compute
the velocity at the end of the interval. Further
fairly large intervals may be assumed provided the
recuperator reaction does not vary greatly at the
limits of the interval.

The velocity curve is computed by this method
to x = bn-d from the out of battery position, where
d = length of the retardation or variable orifice
period, in ft. and b = length of recoil in ft.

The velocity v^ at the end of the acceleration
period is usually taken at approximately 3.5 ft/sec,
at horizontal elevation, though a more rational as-
sumption of the velocity should be based on the fol-
lowing:

Let h = height of bore from ground in ft.
(horizontal c'recoil)

d * length of buffer or variable throttling
interval (ft)

Ws = weight of total system gun + carriage Qbs)

18 = horizontal length from Ws to contact of

wheeled ground (ft)

Cs = factor of stability (» 0.85 usually)
Rh = counter recoil reaction at horizontal

elevation

k = proportional distance of d that the c'recoil
energy is to be dissipated along, k = 0.7 to
0.9

471

Then, the counter recoil reaction at horizontal ele-
vation, becomes,

kd h

With variable recoil, assuming the length of re-
coil to be at short recoil one half of that at long
recoil, in order to have sufficient displacement for
acceleration at maximum elevation the buffer or
variable throttling should not take place at horizon-
tal recoil for over 1/3 to 1/4 the recoil.

Hence d « 0.33 to 0.25 b|,*0.3 bh approx.

/

°'6 C'" *«1'"> - 3.62 /V£i. (ft/,,0)

Knowing vb we may estimate the proper size of the
counter recoil constant orifice. Actually the maximum
velocity of counter recoil is attained shortly after
the out of battery position and at this position the

acceleration is zero. But since the retardation is
very slight until the variable orifice is encountered,
we may assume the recoiling mass to move with uniform
velocity at the entrance to the buffer or variable
throttling. Therefore at horizontal recoil,

=;•*

P. - n W_ - ~

,

W0

Hence, the constant orifice becomes, w(

where v^ * 3.62 /- where for a spring or

"r ^ pneumatic return re-
cuperator system (t s

i b

CQ = -

175

C = reciprocal of orifice contraction factor and
Afc = area of buffer (sq.in)

Fv * recuperator reaction at displacement X»bj,-d (ft)
and for hydro pneumatic recuperator system,

472

--

C0 = • C = reciprocal of orifice con-

traction factor.
Ay = effective area of recuperator

piston (sq.in)

FV » PaAv lbs-

p£ 3 pressure of oil in air cylinder.

For second period of counter recoil: During this

period it is customary to maintain a constant total re-
tarding force which at horizontal elevation becomes,

cov*
P- n 1»-

,
where Rh=cs"~h" — (lbs)

Since the counter recoil reaction is constant
during the retardation, the velocity is a parabolic
function of the displacement, that is

v = 8.03 / - (ft/sec)
"r

Substituting this value of v in the following equation,
we have

(sq.in)

,
b

C A

where for a spring or pneumatic recuperator, Co = •

175

C * reciprocal of orifice contraction factor
Ak = area of buffer (sq.in)

2

Fy = recuperator reaction at displacement x, ^' ^s
for a hydro pneumatic recuperator system, C£ = "" •

C = reciprocal of orifice contraction factor
AT = effective area of recuperator piston
Fv =• pi Ay

pa * pressure in oil in air cylinder (Ibs/sq.in)

473

COUNTER RECOIL With a variable recoil, the re-
FUNCTIONING WITH quirements of proper counter recoil
VARIABLE RECOIL, functioning for all elevations are
more difficult to obtain. At hori-
zontal recoil we must meet the con-
dition of counter recoil stability, whereas at maximum
elevation, the time period of the counter recoil, for
rapid fire, oust not be too long. Since the recoil at
maximum elevation is a fraction of that at horizontal
recoil, the recuperator reaction at the beginning of
counter recoil at maximum elevation is necessarily
smaller than that at horizontal elevation. Further at
maximum elevation we have the weight component resist-
ing motion. Therefore, the accelerating force is
necessarily considerably smaller than at horizontal
elevation and the velocity attained at maximum ele-
vation becomes a function of that at horizontal re-
coil. In the design of a counter recoil system in
order to obtain sufficient velocity in the counter re-
coil at maximum elevation, it is important that a proper
compression ratio be used. This in turn effects the
initial volume of the recuperator and therefore the
entire layout of the recuperator forging. It is here
important to emphasize that proper functioning of
counter recoil can not be attained by increasing pres-
sure where an improper ratio of compression is used.

The following analysis gives a rough approximation
as to the requirements to be met for proper counter
recoil functioning at all elevations with a variable
recoil.

It will be assumed that the recoil at maximum
elevation is reduced to one half that at horizontal
recoil and that a constant orifice is maintained until
the latter third or fourth of the counter recoil. We
have therefore a constant orifice which is the same
for the accelerating period of counter recoil at max-
imum elevation or horizontal recoil.

474

If now

Fyj = initial recuperator reaction

Fvf = final recuperator reaction (Ibs)

Fya) = recuperator reaction at middle of hori-
zontal or long recoil (Ibs)

vs = maximum velocity of counter recoil at
maximum elevation (ft/sec)

vh = maximum velocity of counter recoil at
horizontal elevation (ft/sec)
•Q = area of constant orifice (sq.in)

Co s throttling constant

Rgifp = stuffing + packing friction (Ibs)

0B = maximum elevation

As a first approximation, we will assume,
the maximum horizontal counter recoil velocity
to be attained after a displacement equal to one
half the recoil. Hence

cV

Fvm ~ n Wr - Rs+p - —7 — * 0 (1)

"o

At maximum elevation, the maximum velocity
of counter recoil will be attained somewhat after
a displacement equal to half the recoil, but we
are not greatly in error in assuming the same re-
cuperator reaction Fvin. Hence

<V's
FVB-W (sin0*n cos 0)- RS+D — * 0 (2)

*o
Subtracting (2) from (1). we have

C0<vn-vs>

W-[sin0-(l+ cos 0)nl *
i

hence

C0 Wr[sin0-n(l+ cos0)l

475

We have therefore for required recuperator
reaction at the middle of the recoil

Wr[sin0-n(l+coslB)]

If we assume values for vh and vs for design ap-
proximations, Me may take,vh = 3.5 ft per sec,.

vg = 2.5 ft per sec.
then, FVJB=n Wr+Rs+p+2Wr[ sin0-n(l+ cosfl)]

If we take a large coefficient of guide frict
ion we neglect Rs+p; hence if n - 0.3,
Fvm*0.3 Wr+2Wr[sin0B-0.3(l+ cos00)l

To obtain the minimum allowable ratio of compres-
sion, for spring recuperators, we have 2(?vm-Fyj)=

F ) hence

Ffv • Fvi FvfsFvi+<

Fvf

2(Fvm-0.5Fyi)

m :

Fvi

Fvi

With a pne-umatic or hydropne-umatic recuperat-
or, we have 2.5 (Fvm-Fvi)=Fyf-Fvi (approx.) and

Fyf= Fyi+ 2.5(F9m-¥vi) hence

2.5Fvm- 1.5Fvi 1.5(1.66Fvm-Fvi)

= n =

Fvi Fvi Fvi

RKCUPBRATOBS.

GENERAL CONSIDERATIONS . After the recoil the

recoiling mass must be
brought into battery
and this must take place
at any elevation of the

gun and held there until the next cycle of the
firing. Obviously sufficient potential energy
must be stored during the recoil to overcome the

counter recoil friction and the weight com-
ponent at maximum elevation throughout the count-

476

er recoil. Further in order that the counter re-
coil nay be made in mimimum time, an excess
potential energy is required over that required
for friction and gravity, in order that a rapid
acceleration at the beginning of counter recoil
may be attained. Finally in the battery position
an excess recuperator reaction is necessary over
that for balancing the weight component and over-
coming the friction in case of a slight slipping
back of the piece in the battery position.

Therefore a satisfactory recuperator must
satisfy the following requisites:

(1) The initial recuperator react-
ion should have a marginal excess
over that requirad to balance the
friction in battery and the weight
component at maximum elevation.

(2) The potential energy of the re-
cuperator at the end of recoil
must be sufficient to overcome the
work of friction and gravity at
maximum elevation during the recoil
and rapidly accelerate the gun at
the beginning of counter recoil.

INITIAL RECUPERATOR In general the size or
REACTION. bulk af the recuperator

whether spring or hydro
pneumatic depends upon the
magnitude of t'he initial re-
cuperator reaction. It becomes, therefore, im-
portant to estimate the required initial recuperat-
or reaction to a considerable degree of accuracy.
This is especially true in certain types of recoil
systems where the size of the forging, especially
for guns of high elevation, depends directly upon
the magnitude of the initial recuperator reaction

and it becomes very important to make this a min-
imum.

477

Let Rg = guide friction (Ibs)

R v= packing friction of recuperator (ibs)
Fvi = initial recuperator reaction (Ibs)

ey = distance down from center of gravity of
recoiling parts to line of action of Pv
(in)

Qt » front normal clip reaction (Ibs)
Qf = rear normal clip reaction (Ibs)
x4 and yt - coordinates of front clip reaction

(in)

xa and ya = coordinates of rear clip reaction
(in)

n = coefficient of guide friction = 0.15 approx,

S6m- angle of maximum elevation.

1 = distance between clip reactions (in)
Considering the recoiling mass at maximum elevation
in battery, case of slight slipping back from the
battery position, we must have (see fig )

or

Fyi=n(Q1+Q8)+Wrsin2fm (1)

and normal to the guides, Qz-Qi=Wrcos(? (2)

and taking moments about the center of gravity of

the recoiling parts,

%i *v-Qtxt-Q8xa + nQxyx-n V, = ° (3 >

Substituting (2) in (3), we have,

Fvi ev~Qtxt~ Q2x8~wrcosgfx« * n Qtyt" n a4y, " n

Wf cos/C ya = 0

Fvi ev-Wrcos0(xt+n ya )
hence Qt » - (Ibs) (4)

xt+Vn(y8-yt)
and solving for Q2,

nQfy2 = 0

v a

hence CL - — -

Fvi

478
Hence, with sleeveguides,

*_ "\y*— y* '

I x 9 \

With .grooved guides yt becomes negative and

Since with grooved guides, y^y^ approx.,also
*!+*, * 1 3 distance between clip reactions, and

7tayt * er » lean distance to guide friction, we
have,

v vrBt

Rff » - n (with grooved guides)

2Fvi

r,1a

— — — — — — n (with sleeve guides)

1 (9)

Substituting in eq.(l) we have

t 2f'± 0_+ffrcoi0(x -x..)

- — - - S— *- n +

,

1+2 n e

hence

n cos0_(x -x )
Wr[sineJm+ - " * * ]

1+2 n er
- (Ibs) (10)

2 e..n

l+2n er

and for the initial recuperator reaction,

n cos £L(x -x )
' 3

l+2n e,

^ n

2 ev n
1 - —

1+2 n e,

479

n cot 0B(xt-xt )
1+2 n er

2 ev n

> where k»
1.1 to

1.2 (11)
1+2 n er

1 » distance between clip reactions (in) with 3

clips 1 «-£

2

with 4 clips: 1 » b

b * length of recoil (in)

Estimation of Recuperator Packing Friction 8p:

With hydro pneumatic recuperator systems, the
packing friction is usually a linear function of
the recuperator pressure. Assuming a given in-
itial intensity of pressure pv fflax Ibs/sq.in. in
the recuperator, we have, Rp3Cppv Bax.

The packing friction in the recuperator is
divided into the suffing box friction plus the re-
cuperator piston friction. To estimate these fric-
tions wftmust know the diameter of the recuperator
piston rod and recuperator piston.

To roughly estimate these diameters, we have
for the effective area of the recuperator piston,

1.3Hr(sin£5-+0.3 cos 0ffl)
AT - (sq.in)

PV max

for the required area of the recuperator rod,

2.6Wr(sin0m+0.3 cos 0m)
av « (sq.in)

where fB » allowable fibre stress in rod material.
Then the diameter of the piston, becomes,

(in)
0.7854

480

and the diameter of the rod becomes, d '

v '0.7854

(in)
If wgv = width of stuffing box packing of recuperat-

or(assumed ) (in)
wpv * width of piston packing of recuperator

(assumed Kin )

then assuming the pressure normal to the cylinder
or surface of the rod to be made equal to the hy-
drostatic pressure in the cylinder, we have
Rp-(.06« wpv Dv + .05 n w3v dy)py nax<

- .05n(wpy Dv + wsy dv)py max! (Ibs)

where .05 » approx. coefficient of friction of the
packing.

Approximate Initial Recuperator Reaction:

For preliminary calculations, especially when
the type of packing and arrangement of cylinders
has not been considered we may neglect the re-
cuperator packing friction by increasing the co-
efficient of guide friction.

Without pinching action of the guides in bat-
tery the guide friction, R- » 0.15 Wrcos 0 (approx)
(Ibs). To account, for a possible pinching action,
as well as the packing friction, for elevations
up to 65°, approx. Rg = 0.30 Wr cos 0 (Ibs) and
the required initial recuperator reaction, to al-
low for possible variations, should be increased
from 20* to 30* over that required to hold the gun
in battery. Hence Fyi * 1.3 Wr(sin0m+ 0.3 cos£5)
(Ibs). With guns of very high elevation, Rg =
0.3 cos t becomes negligible. However, the pack-
ing friction remains the same whereas the guide
friction is comparable with that at horizontal re-
coil due to the pinching action of the guides at
maximum elevation. Therefore, it is desirable to
use an approximate formula taking these factors

481

2n
into consideration. We have, appro*. Re =

1

where n = 0.1 to 0.2. If we take n * 0.3 to ac
count for the recuperator packing friction, we
have at high elevations,

0.6 Fvi eb
pvi - 1.3(Wr sin 0m + *Y* ) (Ibs)

where eb * distance from bore to line of action

of Fvj (assumed ) (in)

1 * distance between clip reactions (in)
with 3 clips

with 4 clips 1 = b

b - length of recoil (in)

ENERGY REQUIREMENTS The initial recuperator
FOR PROPER reaction is designed to be

RECUPERATION. somewhat greater than that

required to hold the gun in
battery at maximum elevat-
ion, against the guide and packing frictions.
Further, the recuperator reaction, being necessarily
derived from a potential function, must therefore
increase with the displacement out of battery.
The work done by the recuperator, therefore, is in
excess of that required and we have, always, an
excess potential energy over that required to bring
the gun into battery. This excess energy is dis-
sipated by the counter recoil regulator. We have,
i

therefore, merely a transfer of part of the re-
coil energy, dissipated by Beans of the recuperat-
or, ultimately in the counter recoil. The total
heating or rather the average in a recoil cycle
is quite independent of the magnitude of the com-
pression. However, with high compression ratios,
we have extreme local heating where the radiation
is small and therefore injurious effects are like-
ly to result with the air packings in hydro pneu-

482

•atic recoil systems. Further excessive potential
energy stored in the recuperator, requires care-
ful counter recoil regulation, and as stability
on counter recoil is far more sensitive than on
recoil, we have more difficulty in meeting the
rigid requirements of counter recoil stability.
Finally with excessive recuperator energy to main-
tain low counter recoil regulator or buffer pres-
sures requires a cumbersome and large counter re-
coil regulator whereas it is far simpler con-
structively to dissipate the recoil energy during
the recoil.

Therefore excessive recuperator energy is un-
desirable for the following reasons:

(1) Localized heating resulting with
hydro pneumatic recuperators, is in-
jurious to the packing.

(2) Difficulty in counter recoil
regulation and meeting counter re-
coil stability requirements.

(3) Constructive difficulties due
to a bulky counter recoil buffer or
regulator required to maintain
moderate pressures in the buffer
chamber.

On the other hand, the mean recuperator re-
action must be sufficient not only to balance
the weight component of the recoiling parts and
frictions, but enough to accelerate the recoiling
parts to a given minimum velocity for counter re-
coil at all angles of elevation. Since it is con-
structively complicated and more or less impractical
to introduce varying counter recoil regulation as
the gun elevates in the majority of the types of
recoil systens are designed on the bases of given
maximum velocity at horizontal elevation consist-
ent with counter recoil stability and a given
minimum velocity at maximum elevation, consistent
with reasonable time of counter recoil at maximum

483

elevation. Usually the recoil is shortened at
•aximum elevation. We are not greatly in error
in assuming the respective velocities to be at-
tained at a displacement corresponding to the
oean recuperator reaction, whicb is roughly from
one half to two thirds away from the battery
position.

We have, then, with a variable recoil, if
Pvm = mean recuperator reaction (Ibs)
Rs+p = total packing friction in counter re-
coil (Ibs)

CQ = throttling constant of regulator
w0 » throttling orifice of regulator (sq.in)
vb » velocity of horizontal e 'recoil (ft/sec)
vs = velocity of c'recoil at maximum elevation

(ft/sec)
n - coefficient of guide friction,

for the notion of the recoiling parts at horizont-
al recoil,

<*;

%« - " »r - Rs+p - — = 0

for the motion of the recoiling parts at maximum

elevation,

_ i *
C0vs

FVB -Wr(sinJ0m+n cos 0m)+Rs+p jj— » fr

wo
Subtracting, we obtain

C0 Wrtsin0m- n(l

2

vb

and

X

tfh

FVIB* n

wr+Rs+p +

W_[sin
y» _ va r
vh vs

cos

We see, therefore, that the mean recuperator re-
action required depends greatly on the square of
the horizontal c'recoil velocity and inversely as
the difference between the squares of the borizont-

484

al and maximum elevation, o 'recoil velocities.
Since vn is nore or less fixed by c 'recoil stability
limitations, whereas vg depends upon the time allowed
for counter recoil functioning at maximum elevation,
Fym becomes more or less fixed and therefore the
required excess potential energy of the recuperat-
or.

Assuming design values of vh = 3.5 ft/sec.
and vs = 2.5 ft/sec, with an increased coefficient
of guide friction to compensate for the packing
friction, n = 0.3, we have

Fym = 0.3Wr+2Wr[sin0m-0.3(l+cos 0m)] which gives
a rough approximation as to the value of the mean
recuperator reactions required.

CALCULATION OP THE MEAN RECUPERATOR REACTION
AND THE BNERQY STORED IN THE HBOUPBHATOR.

SPRING RECUPERATORS. With spring return

recuperators, we have
the recuperator re-
action increasing pro-
portionally with the
recoil. If Fvi * SQ = the initial spring re-

cuperator reaction (Ibs)

Fvf * sf = the final spring recuperator re-
action (Ibs)

b - length of recoil (ft)
Then Sf+So Fyi*Fvf

(lbs)

hence Tvf * 2Fym-Fvi (Ibs)

The potential energy stored in the recuperat-
or for s displacement x, becomes

x Sf-S0

W » / (S0+ — - — x)dx

485
c _— c

bf bO 2

= S0 x + - x (ft.lbs)
2b

and the total potential energy required at the end
of recoil, becomes

W = (S0 + Sf)| - (Pyi * Fvf)| (ft.lbs)

With hydro pneumatic or pneumatic recuperat-
ors, we have the recuperator reaction increasing
as an exponential function of the recoil displace-
ment. If

pa a intensity of air pressure in recuperator
at any displacement in the recoil X
(Ibs/sq.ft)

* initial pressure in the recuperator

(Ibs/sq.ft)

* final or maximum pressure in the re-
cuperator (Ibs/sq.ft)

• » ^. - ratio of compression
Pai

Ay * effective area of recuperator piston (sq.

in)
V * volume of recuperator at displacement x

(cu.ft)

VQ = initial volume of recuperator (cu.ft)
Vf = final volume of recuperator (cu.ft)
x = recoil displacement (ft)
b = total length of recoil (ft)
Then,

Pa^ a Pai vo
where k 3 1.1 for oil in contact with air

= 1.2 for oil separated from air by a float-

ing piston.

Since V = VQ - Ayx, for a recoil displacement x,
we have y ^

pa = pai (tr-^T - ^ or ln ternis of tne

° v total recuperator re-
action

486

r

0

O T

Tba work of coapression, becoaes

V V dv

"x » - / P. d 7 » - pai Vj / 21 tft.lbs)

V V Vk

1-k

Since Fy± = PaiAv, we have for the work of com-
pression in terms of the total initial recuperat-
or reaction

Fvi Vo , 1 1 .

wx ' I- 1 k-1) ^yk-i v*-1

*o

where as before V = VQ-Ayx. At the end of recoil,
we have substituting, for V, 7f =

• » — — * £77-) » the ratio of compression.

now

Tbe total work of eoapression in terns of "a" be-
comes - y *;i

Wb - -2 S(H k - 1) (ft. Ibs)

k-1

It is custoaary to measure the pressure in
Ibs. per sq.in. rather than Ibs. per sq. ft. and
the volume in cu. in. The above formulas, be-
come

Pai 7o I 1

12C1C-1J

487

-V —

(B - 1) (ft.lbs) or in terns of

the initial re-

cuperator reaction Fvi and the effective area of
the recuperator piston Av(sq.in) we have

12Av(k-l)

(ft.lbs)

P - V* ~

wb = — — — 2— (• ' - i) (ft. ibs)

12Av(k-l)

v

and PaAv = Fv = pai Av( - - - J (Ibs)

W
where x = recoil displacement (inches)

Ay= effective area of recuperator pistomsq .in;

Vo= initial volume (cu.in)

paj=initial recuperator pressure (Ibs/sq.in)

V = V0-Avx (cu.in)
The mean recuperator reaction, becomes,

Fvivo ***

^m - 1 ) (Ibs) where Ay is in sq.

> ft., b in ft., and

Paf V Vo in Cu'ft'

Since - » m = (--)

Pai Vf

and V* •« — hence

1 m - 1

VQ(1 -- )Avb and Ayb » VQ( - ) therefore

m* at*

1 k-l

Fv« * Fvi(~ - H* "1) (Ibs) which gives the

mean recuperator
reaction in terms

of the initial recuperator reaction and the ratio

of compression

488

Since Pvj * 1.3yfr(sin0a+0.3 cos 0m)(approx.)
(Ibs) we will have

— 0.3Wr( , t)Wr[3in0B-0.3(l-cosg),)]

k-1

1.3Wr(sin0B+0.3 cos0m)

1.3(8in0B+0.3cos0B)

If we assume vn - 3.5 ft/sec, and vs = 2.5 ft/aec,,
then

hence
1_ Jt=i

0.3+2[sin0m-0.3(l-cos0B)]

^T" -)(n^r^
•* - 1

From the above equations, we note that the
proper ratio of compression depends on the angle of
elevation and is entirely independent of the weight
of the recoiling parts. The compression ratio
does depend upon the value assumed for the initial
recuperator reaction, the higher the initial re-
cuperator reaction the lower ratio of compression.
The compression ratio increases with the elevation
for proper functioning of counter recoil at max.
elevation.

If now we construct a table with values of m,
and the corresponding values

489

( ) and a ~ and their product for

1 k-1

k = 1.1 and 1.3 res-
pectively, we >ay de-
termine • by inspection and interpolation, pro-
vided we know the max. angle of elevation. If
we let,

1 k-i

B* - * - 1

A - - ; B •

k-1

r=-T) [ s i n0m-0 . 3 ( 1-c os£fB ) ]

then, where k =1.1
• A

1.3

4.71V

o. 23

1.O84

1.5

3.24'7

0.37

1.201

1.75

2. §08

0.51

1.279

2.00

2. 138

0. 64

1.368

2. 3O

1. 883

0. 78

1. 468

and where k - 1.3

m A

1.3

5.464

.296

1. 129

1.5

3.732

.326

1. 219

1.75

2. 840

.456

1.306

2.0O

2. 420

.577

1.395

2. 30

2. 113

.703

1. 486

490

from the above tables, carves were plotted with
values of C against • for k * 1.1 and 1.3 respect-
ively.

la order to compare the probable velocities
obtained in the counter reeeil at maximum and hori-
zontal recoil for a given ratio of coapression m,
or on the other hand if given values of velocity
at borisontal and naximua elevation are wanted the
following method enables us to determine the proper
value ef the ratio of compression •.

If we plot for various values of m, the cor-

responding value of

for a lean aax. elevation rg at 63°, against vh as
horizontal abscissa and vs as ordinates, we obtain,
a series of curves for the various values of a,
which having decided upon the ratio of compression
to be used enables us to determine immediately the
velocity of c 'recoil at max. elevation for any
given velocity at horizontal recoil.
How, ^

0.3+ - - — [sin*H5.3(l- cos £J)]

1.3(sin0+0.3 cos 0)

0 * angle of elevation. (In this series of cal-

culations, the angle of elevation will be con-

sidered only at 65°).
.-. 0 - 65°.

Sin t - .906308

Cos 0 * .422618

Sin t - 0.3(l-cos0)» .906308 - .3(1 - .422618)* .7330
1.3(sin0+.3cosO)» 1.3(. 906308+ .3 x .422618)«1.343

Various values ef C (equation fl) are given
in table on preceding page.

The only unknown in the equation 111 is the ex-
pression

491

Vu «

vh y"

• Let = K. Taking the various values

Vn~"Vs of C as given in the pre-

ceding table and sub-
stituting in formula #1, we get the following values
of "K", for the given values of "C":

1.1

1.084

1.5*76

1. 201

1.790

1.2-79

1.933

1.368

2.096

1. 468

2.279

1.3

1.129 1.659

1.219 1.824

1.306 1.983

1.395 2.145

1.486 2.312

..-!L

v«-va

h s

Now to show the relation Vh and Vs, a curve
will be plotted for each value of "K" as calculated
and recorded in the table :

Vh - KVn -RVs ; K7! = EV*h - ?J ;

492

KV'U-V!

V.

P
T

Now for each value of K, assume values of Vh, from
0 to 10 and substitute in Formula #2, and obtain various
corresponding values of Vs. These values of Vs
plotted against values of Vn enables us to plot
the curve, the corresponding values of V8 and Vh
for each value of "K".

1.1 SET Or CURVES

M

When K - 1.576 1.3

Vh 123*5678 10

Vg .604 1.21 1.81 2.42 2.79 3*62 4.08 4.84 6.05

When K - 1.790 1.5

^h 12345 673 10
V, .663 1.33 1*98 2.66 3.32 3.96 4.64 5.31 6.64

When K - 1.933 1.75

Vn 12345 6*78 10

V3 .693 1.39 2*08 2.78 3*47 4.16 4.85 5*55 6.93

(Then K » 2.096
Vh 123*5678 10

Vg .72.2 1.44 2.17 2.89 3-62 4.34 5.06 5.78 7.22

When K - 2.279
Vb 12345 67s 10

V, .748 1.50 2.24 2.99 3.74 4.48 5.24 5.99 7.49

493

1.3 BIT or CURVES.

When K » 1.659 1.3

Vn 12345678 10

Vg .63 1.28 1.89 2.52 3.15 3.78 4.41 5.04 6.3

K - 1.824 1.5

Vh 12345678 10

V_ .67 1.34 2.01 2.69 3.36 4.03 4. "70 5.37 7.12

•

K » 1.983 1.75

Vh 12345678 10

Vg .704 1.40 2.06 2.82 3. 52 4.22 4.92 5.63 7.04

K » 2.145 2.00

Vh 1-2345678 10

V8 .734 1.46 2.19 2.92 3.65 4.38 5.11 5.35 7.30

K - 2.312 2.3

Vh 12345678 10

V3 .753 1.50 2.26 3.01 3.76 4.50 5.27 6.02 7.53

SPRING RECUPERATORS. Spiral spring columns, en-
closed in cylinders for pro-
tection, are extensively used
to bring the recoiling parts
back into battery from the out

of battery position. For small guns, spring re-
cuperators are more useful, since they are simple
in construction compact and readily adaptable to
a gun mount. With large guns, however, the
energy required for recuperation is large and there-
fore the spring columns become excessively heavy,
since the weight of the springs is proportional to
the potential energy stored within the springs.

494

495

496

497

Hence for large guns pneumatic recuperators have
become almost universally employed.

The stresses computed in springs are based
merely on their static loading. During the ac-
celeration period of the gun, the spring coils
adjacent to the attachment on the recoiling parts,
necessarily are subjected to a very large ac-
celeration, whereas those coils adjacent to their
attachment on the cradle remain stationary. Due
to the great resilience of a spring column, probably
only a few of the front coils adjacent to the re-
coiling parts are subjected to any material accel-
eration, the spring not being capable of transmit-
ting a force sufficient to accelerate the inner
coils. Due to the very rapid acceleration during
the first part of the powder period we have an im-
pact or very suddenly applied loading on the spring
which induces a compression wave, the peak of the
wave being adjacent to the recoiling parts and the
velocity of which depends upon the inertia per unit
and elastic constant of the spring. It is possible
that some of the failures in the service of re-
cuperator springs are due to the dynamical aspects
of the loading on the springs during the firing.

Since the inertia loading due to the powder
acceleration comes practically on the front series
of coils adjacent to the recoiling parts, the coils
directly adjacent to the recoiling parts become
more greatly compressed and correspondingly stressed.
We should expect the front coils, therefore, to
give the greatest trouble and this has been found
the case in actual service.

Due to the complexity of the problem in actual
calculations of the dynamic stresses in the spring
no attempt will be made here to outline a procedure
for such calculations, and only the static loading
with suitable safety factors based on experience
will be used in the preliminary design of counter

498

recoil springs. Lei

0 = diam. of the helix of the coiled spring

(in)
R * radius of the helix of the coiled spring

(in)

d = diam. of the wire (in)
fs= max. allowable torsional fibre stress

used (Ibs/sq.in)
N 3 torsional modulus of elasticity (Ibs/sq.

in)
T = torque or total torsion at any cross

section of the wire (in. Ibs)
Considering any portion of a spring column
subjected to a conpressive load F (Ibs) , along
the helical axis, we have at any section, through
the wire,

A torsional load T = f R
A shear S = P

If we assume pure torsion at the section, the
torsional fibre stress becoaes,

f » fg — (Ibs/sq.in) where ro = - (in) hence
ro

T » F R - / ° 2*r dr f. —

s T*

o ro

and therefore ,3 a
nfad n fsd

F » = (Ibs)

16R 8D

Next consider the twist of any length of the
wire 1. We have, for the torsional shear displace-
ment of a circumferential annular of the wire,

fs ro

t * — since fa - 0 N hence 0 » r- where 9 « the
N 1

499

angle between two radius of the wire at two sections
1 distance apart. Therefore

a ,£^«!£ii

The relative displacement between the extremities
of the helix lor a load P, producing an extreme
fibre stress f, becomes

2f,Rl

9 » R 9 » — but the length of the total wire

Nd

of the helix, becomes, 1 » 2nRa

approx. * K Dn where n - no. of coils. Bonce
9

We have, therefore, the two fundamental spring
formulas, for springs of circular cross section

Kf d* ufsds

PR = (Ibs) (1)

8D 16R

nf.Dfn
9 » — (in) (2)

B d

The above formulas apply strictly only to
closed coiled springs, no bending being considered;
however, for a first approximation, they may be
used for open coiled springs with sufficient accuracy
for ordinary calculations.

For rectangular wire sections, we have semi-
empiroal formulas for the torsion, and deflections;

* * <«!*? PK >f»

9

.8b
4 o J T 1

where aQ * length of long side of rectangular section

(in)
bQ = length of short side (in)

500

J a the polar moment of inertia of the

rectangle.

1 * length of wire (in)
A * cross section of the wire (sq.in)
No" ab* ba3 ab(a*+b*)

lxx+lyym 12" "12 3 12

Hence for rectangular section spiral springs,
we have,

aobo

fs

~( aobo

} ',

3a0+1.8b0

1

3a0+1.8b0

D

lOnJD'n

— . p
A4 N

166nD'n a0b0(a0+b0)

- -- ta (in) (2»)
A*N (3a0+1.8b0)

If now, we let

a = deflection at assembled or battery height

(in)
b"= displacement somewhat greater than the

length of recoil (in)
Fyi « load at assembled height ) initial re-

cuperator reaction (Ibs)
Fvfl * load at solid height or at deflection

corresponding to (a+b ' )
n * no. of effective coils
N » torsional modulus of elasticity (Ibs/sq.

in)

d * diaa. of wire (in)
D * diai. of helix (in)
H0* solid height of spring (in)
f3= working «ax. fibre stress (Ibs/sq.in)
then:

for circular springs: for rectangular springs:

ab f

501

1.66nD*n

A4 tf " (3aQ*1.8b0fs
(in) (2')

vi

~ '

H0 - nd (4)

In the four equations, above we are given fg
P?9 bf N and Ho

Fve
fa ?„* b ' N D and - —

leaving the four unknowns, d D a and n or d a n and

H0. Therefore a complete solution is possible, and
the proper size spring may be iauaed lately arrived

at.

ENERGY STORED IN SPRING The fibre stress on a

helical spring is direct-
ly proportional to the
axial load, that is

f * — TS- F (Ibs/sq.in) and the corresponding axial
n u _ ns g

deflection, becomes, 9 * f (in) hence the de-

Nd flection of a

helical spring loaded axially is directly pro-
portional to the load, that is

6 = f (in)

Hd4

The potential or resilient energy stored in a
helical spring becomes,

P „ Hd4 at ,.

A » - 9 = — . — 9 (in Ibs)

2 16D»n

If the spring is to be stressed to a maximum
allowable fibre stress fs (Ibs/sq.in) we have

502

n'Dd'n
A » 16 N f*8 (in Ibs)

The volume of the material of the spring equals
approximately

n » n a
7 » - d n * D » - D d n (cu.in)
4 4

Hence the total energy in terns of the volume, is

»f*s

A » - — that is, the energy stored in a spring
4 N

for a given max. allowable fibre stress

and torsional modulus, is directly proportional to
the volume and hence the weight of the spring. Thus
with tbe same maximum stresses and same kind of
material, the weight of the spring is directly pro-
portional to tbe energy absorbed by tbe spring*

The weight of tbe spring in terms of the total
energy stored in the spring, becomes

where Wg * total weight of tbe spring

(Ibs)

ws = weight per cu.in. of tbe
material of tbe spring
(Ibs/cu.in)

RATIO OP COMPRESSION WITH For minimum weight
SPRING RECUPERATORS FOR of a set of counter re-
MINIMUM WEIGHT OP COUNT- coil springs the com-
EB RECOIL SPRINGS. pression ratio is

definitely fixed.

Let Fvi * the initial recuperator reaction (Ibs)
Fvf = tbe final recuperator reaction (Ibs)
Pye = the maximum solid load on the re-
cuperator springs (Ibs)

a * deflection of springs to assembled

height in battery (in)
b * lengtb of recoil (in)
b" * detleotion of springs from assembled to

solid height (in)

503

Since the load on the springs is proportional
to the deflection we have immediately,

Fre a+b* b"

- * - ; and Pra - Fvi(1+ — >

a

The total energy stored in the spring column,

A - - (a+b") = - U+2b' + - — )(in.lbs)
222

Since b" and Pyi are fixed conditions to be not
in the design of the carriage, the only variable
in the above energy expression is a. Therefore,

for minimum weight, w «a

d(a+2b"+ - )

dA a

— s 0 hence

da da

1 * —5— » 0 and therefore a » b"

a

The ratio of compression, becomes,
— — • » 2 » — (approx)

Fortunately this ratio is nearly ideal for
proper recuperation and hence satisfactory de-
signed spring column with minimum weight may be
used .

RECOPERATOR DIMENSIONS With hydro
AND LIMITATIONS. pneumatic recuperators

we have two or more
cylinders, the recuperat-
or cylinder and the air
tank or cylinder. Let

b =» length of recoil (in)

b = corresponding displacement in air cylind-
er

504

Ay = effective area of recuperator piston (sq.

in)
Aa » cross section area of air cylinder (sq.

in)

Paf
m » * ratio of compression

Pai

Aa
r * — - * ratio of recuperator cylinders.

Ay

1 = length of air volume in terms of cross
section area of air cylinder (in)

j » - 3 length of air volume in terns of re-
coil stroke

Vo * initial air volume (cu.in)
Vf * final air volume (cu.in)
Then Vf « VQ-Avb

U U V

P

VQ k

-i£ «

m * (— ) where

k * 1.1 to 1.3

Pai

vf

therefore.

vo

i

vo

*r 3

•

that is Vf «

"T

mk 1

1

,

k

_k

VI

1)=

>ATb hence

V- A V, •

m

0 " , AVb *

i

'Aab'

•" mk- 1 mk -1

which shows clearly, that the initial volume de-
pends only upon the ratio of compression, the
area of the recuperator cylinder and the length
of recoil.

If now, we decrease the effective area of the
recuperator piston, for a given recuperator re-
action, we must increase the intensity of pres-
sure in the recuperator cylinder, that is:

•r M Y\ A

fty| Pyi fly

505

Kvib n
hence VQ * — — 1-- (6)

since pvi » pal » pai approx,

i

Kvib B=
V0 - - i (61)

Pal m* - 1

Now the size of the recuperator depends rough-
ly on the initial volume VA; hence, in pneumatic
or hydro pneumatic systems, it is important to main-
tain as high air pressure as possible.

In recoil systems, where the recuperator and
brake cylinder is one and the same as in the St.
Chamond and Puteaux brakes, the effective area of
the recuperator piston is that of the recoil pis-
ton.

Now the pressure during the recoil is limited
to a given maximum consistent with the packing and
therefore the effective area of the recoil piston
is fixed. With large guns the recuperator reaction
is relatively small as compared with the maximum
recoil pressure, and therefore the intensity of the
air pressure is small. Hence the recuperator volume
and the size of the recuperator is large as compared
with a separate recuperator system, using high re-
cuperator pressure intensities.

Thus for large guns, or guns with low elevation,

separate recuperator systems separate from the brake

system usually gives a smaller recuperator brake
forging.

Limitations of the ratio of compression "m".

The limitations of "m" are fairly
fixed:

(1) The minimum "a" is based on &
consideration of the proper
functioning of counter recoil at all

506

elevations .

(2) The maximum "m" is based on a

consideration of horizontal stability
in the out of battery position for
the recoil, as well as heating and
rise of temperature caused by the
compression of the air.

(1) With guns shooting at high elevation, the
recoil must be shortened for clearance at high
elevations and lengthened for stability at hori-
zontal elevation. Thus high angle guns require a
variable recoil, the ratio of short to long recoil
being usually from one half to two thirds. The re-
cuperator reaction at maximum elevation must be suf-
ficient to bring the gun into battery with a moderate
velocity in order that the time of counter recoil
at maximum elevation may not be too long. This
feature is of considerable importance. Raising the
air pressure in the recuperator, though it will
sufficiently accelerate the gun at maximum elevation,
will give too great a velocity at horizontal recoil
and thus endanger counter recoil stability. Thus
in the initial design it is important that the

initial volume is such that it will give the proper

. .
ratio of compression.

The mean recuperator reaction, or rather the
recuperator reaction at the middle of the recoil,
was shown in the discussion on counter recoil to

be,

t

* v*

s

where vh • the max. velocity a+ horizontal recoil

(ft/sec)

vg - the max. velocity at max. elevation
(ft/sec)
total recoil packing friction

507

B » coefficient of friction from 0.1 to 0.2
For a preliminary design constant, we nay assume,

vb - 3.5 ft/sec. va * 2.5 ft/sec.
and taking a large value of n * 0.3 to compensate

%>-0.31!r+2Wr{sin01|-0.3(l+cos0m)] (Ibs)

With a hydro pneumatic recoil system, ire have

roughly, 2.5(PVB-Pyi)»P>vf-F'vi hence the minimum allowable

ratio of compression, becomes,

*vf 2.5%.-1.5Fvi 1.5(1. 66Fva - Pyi)

m 9 = 3

?vi pvi Fvi

of course the ratio nay be decreased by using lower
values of va or higher values of v^, or both but the
above assumed values give a satisfactory counter
recoil at all elevations.

(2) The maximum value of m is based on the
following considerations:-

(a) Horizontal stability, where
a high final air pressure nay
exceed the allowable overturn-
ing fo^ee consistent with stabil-
ity in the out of battery
position.

(b) The maximum allowable c 'recoil
buffer pressure which linits

the potential energy stored in
the recuperator in the out of
battery position.

(c) The allowable rise of tem-
perature caused by the com-
pression of the air.

With light mobile field carriages , stability
is very often the determining factor for the
maximum allowable ratio of compression. This is
likely to especially occur when the mount
elevates to very high angles and perfect horizontal

508

stability is required as in anti-aircraft material.
If the resistance to recoil consistent with stabil-
ity at horizontal recoil is small, and the initial
recuperator reaction large, a high compression
ratio will cause the total resistance to recoil in
the out of battery position at horizontal recoil
to be greater than the balancing stabilizing moment.

Obviously this critical condition will only
occur with guns of high elevation and required to
meet rigid horizontal stability limitations. In
an ordinary recoil system as it is impossible not
to have more or less throttling at the end of re-
coil, we must have the maximum allowable re-
cuperator reaction a fraction of the total pull for
minimum elevation of stability 0j.

Therefore the maximum allowable ratio of com-
pression from a stability consideration, becomes

fvf 0.8[Kh+Wr(sin0i-0.3cos0i)]
mmax * •=— » (min. elev.)

0.8(Rh-0.3Wr) Kh
» "» = 0.75 — — (horizontal

Fvi Fvi elevation)

Therefore, when Fvi is large and Kh small as
with guns for high elevation and rigid stability
requirements "a" becomes small and low ratio of
compressions with corresponding larger recuperators
are required. Very often in anti-aircraft material
"m" becomes smaller than that required for proper
counter recoil functioning at max. elevation. In
such a case it is preferable to sacrifice horizontal
stability somewhat and increase the horizontal re-
sistance to recoil.

The previous formula may be expressed direct-
ly in terns of stability. If

Ws * weight of the total mount (Ibs)

509

13 = horizontal distance from spade to Wg

(ft)
bh = length of recoil at horizontal elev.

(ft)
£.£ * min. angle of elevation.

We have for the max. compression ratio based on
stability,

0.8[(Wgls-Wr bbcosefi)

m =

M

and at horizontal recoil,
0.8[Wsls-Wr(bh-0.3))

,

0.75 (

rvi

In counter recoil systems using some form of
a c 'recoil regulator of a buffer type, we bave a
necessary geometrical limitation in the maximum area
of the buffer. Thus in filling in types of buffers
as in the Schneider and Filloux recoil systems as
well as ordinary spear buffers which enter the pis-
ton rod at the end of c 'recoil, the effective
area must necessarily be considerably less than the
area of the piston rod.

With spear buffers attached to the piston due
to void considerations at the beginning of the re-
recoil, we again are limited in a large effective
buffer area. If

Pb max * the max. average allowable buffer
pressure (Ibs/sq.in)

b = length of recoil (ft)

Ab » effective area of buffer

db = length of buffer c 'recoil (ft)

Rp ' total packing friction (Ibs)

WQ * total potential energy of the recuperator

(ft. Ibs)

then, when the counter recoil brake comes into
action towards the end of c 'recoil, as with a spear
buffer, we have,

510

Pb aax * " and where the counter

Abdb recoil brake is ef-
fective throughout the counter recoil,

•0-(Wpsin0+Rp) f

Pb max " ' ~^- "h«re V

• * ratio of compression

?o • initial volume of recuperator (cu.ft)

Fvi - initial recuperator reaction (Ibs)

k * 1.1 or 1.3 depending whether air is in contact

with oil or separated from it by a floating

piston.

The expansions for pD assume a constant buffer
fere* during the buffer action. This however is
not always the case and therefore the above ex-
pressions should be multiplied by a suitable
constant to take care of the peak in the buffer
pressure when the buffer pressure is not constant.

It is to be particularly noted that the peak
buffer pressure may greatly exceed the average
buffer pressure as obtained by the above expressions

Combining the above expressions, we have

.ax

, „
This is a very important limitation for m and is

inherent for all direct acting counter recoil buffer
brakes. Values of pb max range as high as 8000 to
10000 Ibs/sq.in. with short spear buffers but such
pressures should not be tolerated on future designs.

In general the c 'recoil buffer pressure
should be maintained as low as possible, thus sim-
plifying the design of a counter recoil system;
therefore, the lower value of m consistent with a
satisfactory functioning of c 'recoil at all
elevations should be used.
(3) Though the total energy dissipated in a re-

511

coil cycia Bust necessarily equal the initial recoil
energy, it is important to distribute the energy
in the parts of the system where radiation is most
effective, if the energy is dissipated entirely
in the throttling both on recoil and counter recoil
ire have a large nass of oil with corresponding radi-
ating surface. With high compression ratios the
air in the recuperator rises to a high temperature,
which nay cause injury to the packing and lubrication,
and therefore it is important to Maintain a low com-
pression ratio and thus decrease the localized heat-
ing in the recuperator where radiation is the
smallest 4

As to the allowable rise of temperature to be
permitted, depends greatly upon the type of pack-
ing to be used and the packing specification should
state the allowable temperature rise.

Tne temperature T at the end of a recoil stroke,
above the mean temperature Tm at the beginning of
the stroke, may be obtained, from the relation,

T. Pai

Assuming a ratio 2, and a mean temperature 25°
centigrade, we have

T =» 298 x 2°*23 = 349°, when k « 1.3 and therefore
the rise of temperature becomes, T-Tm=51°C or 92°F.

The temperature rise increases considerably
with the ratio •, thus when • « 2.5, T - TB » ?a°C
or ISff'F.

RECDPERATOR DIMENSIONS With hydro pneumatic
AND LIMITATIONS. recuperators we have

two or more cylinders,

the recuperator cylinder and the air tank or
cylinder.

512

Let b = length of recoil

Ay = effective area of recuperator piston
Aa » cross section area of air cylinder

Paf
m = — ratio of compression

Pai

Aa
r = — j[- = ratio of recuperator cylinders.

1 = length of air volume in terms of cross
section area of air cylinder

j = - a length of air volume in terms of recoil

b stroke.
Then, the initial volume becomes,

nk Aa 1

V0 x A_l = A_b but since — » r: - s j

- Av

,k - l V

m

I

i

rj

hence k

i

M* -1

When a floating piston separates the oil and air,
k = 1.3 (approx.) Whan the oil is constant with
the air, k * 1.1 (approx.)

§ ^a _ 1 •*

' * j i

• k -1

1 i

k

•k (r-1) - r

i

k r

513

Tables for a and r for various air column lengths
when k = 1.3 are given below:

r

r

r-

•1.66

-1.66

log

1.3 log

•

r

3

i.

34

2.

239

. 35005

. 4550*7

2.

851

3.5

i.

84

1.

902

• 2*7921

•36297

2.

307

4.

2.

34

1.

209

. 232*74

• 30256

2.

007

4.5

2.

84

1.

585

• 2O003

• 26004

1.

820

5.

3.

34

1.

497

. 1*7522

.22779

1.

69O

5.5

3.

84

1.

432

•15594

.20272

1.

595

6.

4.

34

1.

382

.14051

.18266

1.

523

r

r

r-

1.25

-1.25

log

1.3 log

•

r

3

i.

"75

i.

"714

.23401

.30421

2.

015

3.5

2.

25

i.

556

. 142O1

.24961

1.

777

4.O

2.

•75

1.

455

.16286

.21172

1.

628

4.5

3.

25

i.

385

. 14145

.18389

1.

527

5.

3.

•75

1.

333

. 12483

.16228

1.

453

5.5

4.

25

1.

294

.11193

.14851

1.

398

c.

4.

75

1.

263

. 10140

.13182

1.

355

514

r

r-1

r
(r-1)

log^T

1.31 log

^1 "

3.00

2.00

1.5000

. 17609

. 22892

1.694

3.50

2. 50

1.4000

. 14613

.18997

1.549

4. 00

3.00

1.3333

. 12483

. 16228

1.453

4. 25

3.25

1.3077

. 11611

.15094

1.416

4.50

3.50

1. 2857

. 1O924

. 14201

1.387

4.75

3.75

1.2667

. 10278

.13361

1.360

5.oo

4.00

1. 2500

.09691

.12598

1.337

5.50

4. gO

1. 2222

.os7o7

.11319

1. 298

6.00

5.00

1. 2000

.07918

.10293

1.267

3

2. 156

1.391

.14333

. 18633

1.536

3.5

2. 656

1.318

. 11992

.15590

1.432

4.O

3.156

1.267

. 10278

.13361

1.360

4.5

3.656

1.231

.09026

.11734

1.310

5*0

4.156

1.20}

.08027

.10435

1.271

5.5

4. 656

1.181

.07225

.09393

1.242

6.

5.156

1.164

.06595

.08574

1.218

515

516

517

r r-.71!

5 --.715

log

1.3 log

i

r

3. 2.285

1.313

. 11826

.15374

1.425

3.5 2.785

1.257

.09934

.12914

1.349

4.0 3.285

1. 218

.09565

.11135

1. 292

4.5 3.785

1. 189

.07518

.09773

1. 252

5.0 4. 285

1.167

,o67o7

.08719

1.222

5.5 4.785

1. 149

.06032

.07842

1.198

6. 5.285

1.135

.05500

.07150

1.179

Vo~ lAvb

b

'.- 7 M

pvdV = |nr

•Vjj+WpCsin^

nax+u cos2fBax

1U4A HI a A

'!-e

where VB *

2 ft/sec,

i

roughly.

Now pvVk»pvi

v|;

fcence py = pvi Vk — k
hence

PV1 ^ V0-.375A¥b 7k

where k = 1.1 to
1.3

- (V0-.375 A.

B

1-k

The solution of this expression is com-
plicated and trial values of VQ nay be substituted
more easily. _ ,

Knowing VQ and Vf = VQ-Ayb, we have •

V

Values of m greater than this value are en-
tirely unnecessary for satisfactory functioning »t
counter recoil at all elevations. When the initial
value of the recuperator reaction is »ade greater

518

than thai required to bold tba gun in battery, the
necessary ratio of m decreases in tbe limit if m »
1, then

*

Kvi*PviAv

4mrv

Due to the uncertainty and variation of both
packing and guide friction, an excess initial re-
cuperator reaction is always used and thus even for
very low values of "•" we usually have in modern
artillery a surplus of potential energy in tbe re-
cuperator.

glilRAL DlSiaH LIMITATIONS.

SURVEY OF LIMITATIONS The design limitations
IN CARRIAGE DESIGN. for a gun mount depend

primarily of course on the
? . particular use to be obtained

from the gun and the general

type of carriage to be used. Though each design
la a problem by itself, it is however possible to
derive and point out certain broad limitations
that lust be observed for a satisfactory design.

The fundamental requirements and limitations
for the various classes of mounts are considerably
different. The question of elevating, traversing,
etc. certain more strictly to a given mount. Row-
tVer, certain broad limitations apply to tbe various

olaases of mounts and for good design these limit-
ations must be always considered quite independent
of tbe requirements for the particular service of
the gun.

(1) For mobile mounts minimum weight
and stability under firing conditions
are primary limitations.

(2) For caterpillar mounts minimum
weight and stability under firing

519

conditions are again primary
limitations.

(3) For railway mounts, due to size
and cost of parts, minimum weight
consistent with stability is ia-
portant but other factors such as
clearance, method of loading, etc.
have perhaps more influence on the
design.

(4) For stationary mounts for defense
work stability is easily secured and
though it is highly desirable to keep
the size and weight of parts as small
as possible, the vital factors are
accessibility, ease in loading and
endurance.

LENGTH OF RECOIL The strength of a gun car-
AT HA XI MUM ELEVATION riage depends roughly on the
AND MAXIMUM RECOIL maximum recoil reaction.
REACTION. How the recoil reaction varies

roughly inversely as the

length of the recoil for a given recoiling mass and
ballistics; therefore it is highly desirable, for
lower stresses in the carriage, to maintain as long
a recoil as possible. But at maximum elevation
we are immediately limited by clearance *of the gun
striking the ground or platform. As the height of
the trunnions and axis of the bore are fixed by
stability at horizontal elevation clearance in
traveling and accessibility for loading, the recoil
at maximum elevation (as well as the maximum recoil
reaction) becomes definitely limited.

Means for increasing the recoil and thereby
diminishing the recoil reaction are as fellows:

(1) By digging a pit under the gun.
(2) By placing the trunnions as far

520

as possible to the rear adjacent to
the breech end of the gun and balanc-
ing the tipping parts by the use of
a balancing gear.

(3) By raising the trunnions as the
gun elevates, obtaining a low height
of the trunnions above the ground
when stability is required and a
high position when stability is no
longer a requirement and a long re-
coil is desired.

LENGTH OP RECOIL AT As mentioned before,
MINIMUM ELEVATION howitzers are designed for
STABILITY. high angle fire, ranging

roughly from 20 to 70 de-
grees. Therefore, stability

is not of great importance up to 20 degrees ele-
vation. At this elevation the moment arm of the
overturning force , about the trail support be-
comes small, and therefore it is possible to con-
siderably raise the trunnion and thereby lengthen
the recoil at maximum elevation than with guns.
Further for a given height of trunnions" the length
of recoil can be shortened for an elevation of 20°
consistent with stability. Thus with howitzers,
it is possible to maintain a constant recoil length
for all elevations. This is of more or less ad-
vantage in simplifying the recoil system.

With a gun, the elevation ranges roughly from
0° to 50°. At 0° elevation the overturning moment
about the spade support is a maximum, and the
stabilizing moment a minimum. (See Chapter III).
Hence a long recoil is, essential in order to reduce
the recoil reaction and overturning moment.

The maximum horizontal recoil however is
limited, due to the fact that at the end of recoil

521

though the overturning moment is decreased by
lengthening the recoil, the stability moment is
also decreased in the out of battery position due
to the recoiling mass being displaced to the rear.
Thus we arrive at an initial length of recoil
where further increase causes a decreased stability.
If Hs = weight of carriage and mount together

(Ibs)

Rh = horizontal recoil reaction (Ibs)
Vf * max. velocity of free recoil (ft/sec)
wr = weight of recoiling parts (Ibs)
b » height of axis of bore above ground (ft)

Then w

0.47 "r T»
Rb * — ~ T~ M (approx.).

then Rh h + Wr b = Hgls at critical stability. Now
the actual overturning moment, becomes,

. ,fr b and the corresponding stability
b g

moment = "s^s

If we differentiate the actual overturning
moment with respect to b and equate to zero, we ob-
tain, the maximum allowable horizontal recoil for
a given recoiling weight, hence

0.47W.VJ h 0.47W.VJ h
d( £-1— + w b) , L£_ + Wp » 0

d b b*g

hence bb max - 0.121 Vf /~h~

Another limitation on the length of recoil
at horizontal elevation, is due to the fact that
as the recoil lengthens, the distance between
the clip reactions decreases, and the clip re-
actions and the guide frictions become excessive
in the out of battery position due to the over-
hanging weight of the recoiling parts. Such ex-

0.471»r V h

522

cessive guide friction caused by the moment of the
overhanging weight combined with the recuperator
reaction at the beginning of counter recoil may
prevent satisfactory return into battery,

Further the bending moment at the rear clip
reaction of the gun becomes excessive due to the
large overhang in addition to the recoil pull on
the gun lug. Thus the length of horizontal recoil
is limited by the minimum allowable distance between
clip reactions when the gun is out of battery. If,
with this maximum recoil the mount is unstable,
either the weight of the mount oust be increased
or outriggers reaching further out must be used.
But for mobile mounts minimum weight is essential,
hence extended outriggers or increase of trail
length must be resorted to. As the gun elevates,
stability increases and the recoil may be shortened
consistent with clearance and stability.

Kith anti-aircraft guns, it is desirable to
shoot from 0° to 80° since the piece must be inter-
changeable for field work if necessary. Therefore
the limitations on anti-aircraft material are more
pronounced and the change of length of recoil is
greater from 0° to max. elevation than with other
types of mounts.

RECOILING WEIGHT FOR The weight of carriage
MINIMUM WEIGHT OF GUN proper not including the
CARRIAGE. recoiling mass is more or

less proportional to the
necessary strength re-
quired in the carriage. Now the strength of the
carriage is roughly proportional to the maximum
recoil reaction. Further the weight of a car-
riage depends upon the type or configuration of
the mount. Hence, for any given type of car-
riage the weight is roughly proportional to the
maximum recoil reaction. If, therefore, a given

523

type of carriage is designed to withstand a given
recoil reaction, the higher the carriage is stressed,
the smaller becomes the ratio of the weight of the
carriage to the recoil reaction. Therefore the
weight of efficiency for a particular type of mount
is increased by decreasing the weight of the mount
per given length of recoil. Obviously if a given
type of mount was designed so that all its parts
were stressed to the elastic limit for the
maximum recoil reaction we would have the minimum
possible weight for the given type of carriage.
Let wc - weight of the carriage mount not including

the recoiling mass.
R * the maximum recoil reaction.
c= weight of the carriage mount proper when

stressed to the elastic limit,
k = the weight constant for the carriage mount

proper.

k1 * the weight constant when stressed to the
elastic limit.

Then '

c > i ' c

k • r and k ' r

Obviously the weight efficiency in a given design
pertaining to a given type of mount, becomes,

k "c
weight off. * — 3 —

k' Wg

Now the weight efficiency varies considerably
with the type of carriage used, certain types hav-
ing considerably more dead weight than other types.
Further the weight efficiency depends directly on
the factor of safety recommended in the design.

A table for the constant "k" for various types
of mounts is given below:

524

Weight

Re.

Max.

Length

Weight

Weight

of

coil-

Recoil

of Re-

of

Con-

Sy«-

ing

React-

coil.

Mount

stant

te«.

wt .

ion.

not in-

of

clud-

Car-

ing

riage.

Recoil-

ing Wt.

Carriage

W.

»r

R

L

We

K

3'Model of

2520

960

4923

45

1560

.317

1902.

75 -/»

265*7

1050

5250

49

1607

.306

Tr e no h

M. 189*7.

75«/» v.of

3045

911

12100

46

2134

.176

1916.

and

18

3.3" g«n

43*72

1435

2100

45

2937

.146

Carriage.

and

30

3 . 8 " How. C «r-

2040

935

13750

4O

1105

.08

r iage, 1915.

a ad

22

4.7" Sun,

7420

2745

17500

70

5675

.324

v. 1906.

4. 7'How.O»r-

3988

1372

19430

52

2616

.135

r iage, 1908.

and

24

155«/m He*.

7600

3498

390OO

51.4

4100

. 105

8 o ha.

155-/-1' il-

19860

9050

66000

43

10810

.164

leaz .

and

71

8" Vioker«,

2OO48

9356

11730O

52

1O692

.091

Mk. Til.

and

24

24O •/•

41296

15790

15OOOO

46.

25526

.171

8CHIIIDBR.

To give a farther physical conception of the
meaning of "k" we note from previous calculations
that the 155 m/m Filloux is extra strong, most of
the fibre stresses not exceeding 10,000 Ibs. per
sq. in. Comparing it with the 3.3 inch, a somewhat
similar type of mount we would expect the 3.3 inch
to be well stressed. This is actually the case.
Two very similar types of heavy field trail car-
riages are the 8" dickers and 155 m/n Schneider,
both having the same type of trail. Both car-
riages are well designed, having in the various parts
about the same maximum fibre stress. Therefore as
we would expect the constant "k* is approximately
the same. The 3" Model 1902 is not efficiently de-
signed as compared with similar types such as the
75 m/m M.1916. We thus see that "k" when compared
with types of similar carriages gives us a crude
idea as to the efficiency of the design of the
carriage itself.

Now the weight of the system is the recoiling
weight plus the weight of the mount proper (i. e.
the stationary parts), that is wg » wr + wc

where ws = the weight of system
wr =* recoiling weight
we = weight of stationary parts, or mount

proper.

Per a given type of mount, the weight of
carriage may be assumed roughly proportional to the
recoil reaction, that is, wc = k R

Now from the principle of linear momentum,
neglecting the small effect of the recoil reaction
during the powder pressure period, and the air
resistance, then m v + I 4700 » m ? wnere m and v -
mass and muzzle vel. of projectile.

I 4700 = the momentum effect of the powder
gases, hence + -
V =

526

but R * — — approximately.
26

(•v+I 4700)*

2»rb

k u1 <<•**• 4700)'

hence R » — where k * -

therefore wft s k R

2b

k k '

Now for minimum weight of the total system,
recoiling parts together with carriage mount,

dw d(wr+-jU^)

s "r kk

r — » 0 that ia - - 1 - •— » 0

d"r i«r »*.

w*-kk'

or w_ =

•r

"ft

where It * s" obtained from table

Jot)

ballistic constant

To use the above fornula ia a new design we
take the value of k fren a siailar well designed
type of carriage, using a somewhat lower value of
"k" according to the judgment of a designer IB
improving the weight efficiency of the mount proper
over a similar previous design. Knowing the

ballistics of the new mount, we find a very definite

weight for the recoiling mass.

It is interesting to note that usually the

strength curve of a gun say be considerably increased

if the proper weight of recoiling mass consistent

with minimum weight is used.

C H A P T B B VIII.

This chapter contains a discussion of
some of the types of hydro-pneumatic recoil
systems with calculations of characteristics of
service designs.

It has been found desirable to print
this chapter separately.

527

CHAPTER IX.

HYDRO-PNEUMATIC RECOIL SYSTEMS.
(Continued)

SCHNEIDER RECOIL The Schneider recoii
SYSTEM. system consists of an in-

dependent recuperator sys-
tem of a hydro-pneumatic
type. The cylinders are

in one forging and are secured to the gun. The
cylinder forging is known as the sleigh or slide
and recoils with the gun. The brake and recuperat-
or rods are held stationary and attached at their
ends to a yoke on the cradle. The hydraulic
brake piston rod is hollow and contains a filling
in buffer chamber. Attached to the sleigh and
sliding within this buffer chamber is a counter
recoil buffer rod. The throttling during the re-
coil is effected through an orifice formed by the
difference in areas of a circular hole in the pis-
ton and the area of the buffer rod. For varying
the throttling, the areas of the buffer rod are
tapered, i. e. the diameter of the buffer rod
varies along the recoil.

The recuperator cylinder consists merely of
the stationary recuperator piston which moves
relative to the forging on recoil. The recuperat-
or cylinder communicates by a large passage way to
the air cylinder partly filled with air. The air
cylinder is placed forward and is made shorter
than the recuperator and brake cylinder. This is
necessary in order that at maximum elevation the
oil in the air cylinder covers the passage way
communicating with the recuperator and air
cylinders. It is very important in the initial

529

530

•

531

lay out of the Schneider recuperator system that
at a maximum elevation the oil completely covers
the communicating passage way in the air cylinder
and the recuperator initial volume should be
reckoned in the air tank beyond this oil cover-
ing. The passage way is made sufficiently large
so that we have practically no throttling in the
recuperator system.

During the recoil, figure ( I ), the brake
throttling is effected primarily through an
orifice formed by the counter recoil rod in a
circular hole in the piston. The simultaneous
compression of the air recuperator during the re-
coil takes place practically along an isothermal
curve, due to the fact that oil and air are in
direct contact in the recuperator. It has been
found by careful computation, however, that an ex-
ponent equal to 1.1 gives a close approximation
in the compression curve of the air and the com-
pression of recoil in the brake cylinder. The
buffer is filled by the pressure head in the re-
coil cylinder, the oil passing through fairly
large orifices in the buffer head FF, the slide
of the buffer head being away from the counter
recoil buffer rod, see figure ( I ).

During the counter recoil the slide on the
buffer bead is pushed in contact with the buffer
rod, and the apertures which filled the buffer

chamber during the recoil are thereby closed and
the throttling now takes place through new orifices
of a very small magnitude. The buffer chamber
having been completely filled during the recoil
enables us to have a continuous regulation through-
out counter recoil. The counter recoil throttling
is effected through a constant orifice for over
half of the counter recoil. We then have a taper-
ing orifice until the gun nearly reaches the in
battery position.

532

In the Schneider system the recoil is designed
constant at all elevations or practically so, a
slight variation taking place with the elevation.
The recoil system is made to vary according to the
stability slope at the minimum firing angle of ele-
vation.

The primary advantages of the Schneider sys-
tem are:

(1) An increased recoiling mass due
to the recuperator sleigh contain-
ing the cylinders, recoiling with
the gun and thereby decreasing the
reaction on the carriage.

(2) The simplicity of the recoil
mechanism, especially from a
fabrication point of view.

The disadvantages of the Schneider system, are:

(1) due to the fact that the primary
element of simplicity, the throttling
effected through a simple tapering
counter recoil rod, inherently pre-
vents any possibility of a variable

recoil.

•

(2) the massive sleigh or slide at-
tached to the gun, though reducing
the reaction on the carriage, lowers
the center of gravity of the recoil-
ing parts below the axis of the bore
so that on firing a large load is
thrown on the elevating arc. To off-
set this, on snail caliber guns a
counter weight has been mounted on
top of the guns. On the larger
caliber guns as in the 240 m/m
howitzer, a brake clutch was intro-
duced on the shaft of the elevating
pinion which slipped during firing.

533

534

Further tba air cylinder, "being necessarily
placed forward of tbe recuperator brake cylinders
with a long recoil gun requires a very long forg-
ing and corresponding guides on the cradle.

On tbe wbole the Schneider recoil system has
proved one of the most satisfactory recoil systems
used during the late war, being simple to fabricate
and tborougbly rugged, due to its simplicity in de-
sign.

Example and calculation of tbe Schneider recoil
system for the 240 m/m Howitzer: As an example of
a satisfactory recuperator brake especially adaptable
for a howitzer, calculations in tbe design layout
of tbe 240 m/m howitzer recoil system are given in
tbe following:-

BIOOIL CALCULATIONS 240 M/M SCHMKIDBR HOWITZER.

Type of gun - 240 m/m howitzer

Total weigbt at recoiling mass * 15,790 Ibs. * Kr

Muzzle velocity - 1700 ft/sec. * Vm

Length of recoil B" 44,833 46,73

Angle of elevation 0 10° 60°

-i pa
Intensity initial air pressure Pal » -— * 576

•7854 "a ik./
Initial air pressure - .7854PaiD* »18800 Ibs. sq.in,

Height of axis of bore from ground— 43"

w V*

Mean constant pressure Pa * » 1,189 x 10 Ibs.

64.40

Weight of powder charge if 40 lb«.

Travel of projectile in bore - u - 160"

Maximum powder pressure on base of projectile Pm =
2005 x 10' Ibs.

535

Maximum pressure on breech Pb » 1.12 Pm

Initial air volume V° = 2970 cu.in.

Final air volume Vf = V° cu. in - Ae b" = 1510

Vi t

Final or maximum air pressure » p.* * pa4 ( — )

Vf

INTERIOR BALLISTICS.

e =' twice abscissa at maximum pressure

D

- i ±
16 Pe 16 Pe

muzzle pressure on base of breech

£Z.e* —^ m pfe 622,000

4 (e+u)*

Velocity of free recoil
w V + 4700 if

50.25ft.sec.

Velocity of free recoil - projectile leaving muzzle

w Vm + .5w V_
yo , » 2. 40.15ft.sec.

Time of projectile to muzzle

t , i-JiL

* * 12 Vm

.01175 sec.

536

Time of expansion at free gases

2(Vf-V0 ) Wr

t » « .01538 sec

P0b 32.2

Free movement of gun while shot travels to
muzzle

* 12(Wr+w+i)

Free movement of gun during powder expansion

pob «<•*
Xa » — — - + V0t8 .7179ft.

Wr 3

Total free movement of gun during powder
pressure period

Z = Xt + X, 1.0279

Time of pressure period

T » tt + t^ .02713 sec.

Total resistance to recoil in battery

mrvj + m(b-E)a
K « ' • " ' Variable recoil

m T*

2[b-E +VfT- - — (b-E)]
2 mr

where K * total resistance to recoil during powder
period (Ibs)

b * length of recoil (ft)
E * free displacement of recoil during pow-
der period (ft)

537

T = total powder period (sec)

c »r

m = cos 0 stability slope

d

c = constant of stability

d = distance from line through center of
gravity of recoiling parts parallel to
bore to center of pressure exerted on
spades.

, 490 - f

g 32.2

C \fr cos 0 cwr 0.85 x 15780

m = - - - = — (approx. ) = - — - -

d h <j . oo

« >::*-f.-iM»iri-!^ f **.-:• "rV- -*•
3760

E = 1.0279 ft.
T - .02713 sec.

b = 44'833 » 3.736 ft.
12

Hence

490 x 50755* + 3760(2.708)*

K

3760 .02713*

2 (2. 708+50. 25x. 02713 x x 2.708)

2 490

1264660

8.13

155000 Ibs. (approx)

Total resistance to recoil out of battery

Rt8
k - K - m(b- E + — )

2mr

» 155000 - 3760(3.736-1.028+ 15500° x -0271 )

2 x 490

538

» 155000 - 3760 x 2.824 » 144,000 Ibs.

CALCULATION OF THE VARIATIOH OF TH1 HI-
ACTIOB AIB PRESSURE IK TH1 HBCOIL.

Initial air volume - 2970 cu.in.

Initial air pressure = 576 Ibs/sq.in)

Length of recoil (10° elevation) = 44.8 inches.

Length of recoil (60°elevation)»46.73

Effective area of recuperator piston = 35.766 Ibs.

Effective area of hydraulic piston = 31.2

Final Pressure (Initial volume)

___^ .^. _ — — — — »___^_——,

Initial pressure (Final volume)

Final volume » initial volume * area at recuperat-
or piston x length of recoil.

.-. Final pressure (10° elevation)- 576( - - )*•*

2970-35.766x44.8

' = 576 x 2.345 - 1350 Ibs/sq.in
1368

Final pressure (60° elevation)
2 970

576(

2970 - 35.766 x 46.73

,,2970.1.1

- 576 x 2.49 - 1434 Ibs/sq.in.
1.299

For 40" Recoil

2970 * " *

Final pressure * 576( )

2970-35.766 x 40

576 x 2.065 * 1189 Ibs/sq.in.

539

1389 x 35.766 » 42525 (Plot these values above fric-
tion)

For 35" Recoil

Final pressure * 576 (- )

2970 - 35.766 x 35

576 * 1.815 = 1045 Ibs/sq.in.
1045 « 35.766 « 37375

For 30" Recoil 2g?Q

Final pressure » 576 ( )

2970 - 35.766 * 30

576 x 1.643 » 946 Ibs/sqiln.
946 x 35.766 - 33835

For 25" Recoil 2970 *•*

Final pressure » 576 ( )

2970 - 35.766 x 25

576 x 1.483 - 854 Ibs/sq.in.
854 x 35.766 = 30544

For 200 Recoll ^^

Final pressure * 576 ( )

2970 - 35.766 x 20

576 x 1.35 » 788 Ibs/sq.in.
778 x 35.766 » 27825

For 15- Recoil ^

Final pressure » 576( )

2970 - 35.766 x 15

576 x 1.22 * 702 Ibs/sq.in.
702 x 35.766 « 25107

For 10" Recoil

Final pressure « 576( - )

2970 - 35.766 x 10

576 x 1.155 = 665 Ibs/sq.in.
665 x 35.766 « 23784

540

For 5" Recoil 2970 *•*

Final pressure « 576 ( )

2970 - 35.766 * 5

576 * 1.072 » 617 Ibs/sq.in.
617 x 35.766 - 22067

Calculation of Velocity Curve

(During Powder Pressure Period)
Point #1. Coordinates Vo and Xo

Vo ' V£o

Kt

o

m

r

Kt

l«o's;

When the projectile leaves the

muzzle,

K = 155000 total resistance to

s

recoil

r

u * travel of projectile in bore

160"
vo= muzzle velocity = 1700ftsec.

w » weight of shell = 353

w = weight of powder charge * 40
Wr * weight of recoiling parts

15790

m - - = 490
32

(w * .5w)V0 3 u

— - - ; t0-^-

(353 + .5 « 40 x 1700)

=40-15

3<16° : .01176

2 x 12 x 1700

... ,0.-40.1S-155000> -01175. 40.15-3.70!

490

36.449ftsec.

541

w + .5if (353 + .5 * 40)160

Xfo * — w - u s -

15790 * 12

155000 x .01175
"2 x 490~

.32 - .0221 = .2979 ft.= 3.57 inches.

Point #2.

Maximum restrained recoil velocity and correspond-
ing orifice.

T *

K(T-tQ)

tm » .02713 -

.01175)

155000 (.02713-
622000

tm » .02329

Vfn » 40.15 * 6220°° [.02329 - .01175]!-
490

62000(. 02524 - . 01175)
4x490(50.25-40.15)

40.15+9.328

v = 49.478
f m

s 49.478 . 15500X.02329
m 490

.see.

Xfm " 2m.

542

fo + — (*•-*(>> - , >
mr 6mr(Vf-V0)

Xf0 * — : — x u » .32 (see Point II)

.32+[40.15+6220°° (.02329-. 01175)-
490

(.02329 - .01175)'

6x490(50.25-40.15)

= .32*. 632 = .952 Ft* 11.42 in.

155000 * .02329*

X. » .952 — = .952 - .0855 »

2 x 490

.866ft. -10. 39 in.
Point *3 (At end of the powder period)

155000 » .02713

490

Vr » 4l.7ft.sec.

X

r 155000 x . 02713*

Xr »

2 x 490
1.0279 - .1155 - .9124ft« 10.94in.

Velocity Curve (during retardation period)

/2CK- •"- (b+X-2Xr)0>-x)]
Vx « /

mr

For x » 1.5 feet.

543

3760
[155000 — <3. 89+1. 0279-2x. 9124)] (3.89-1.5)

•

490

37.4 ft. per sec.
For x » 2 feet

3760
21155000 (3. 89+1. 0279-2 x. 9124)] (3.89-2)

P.

490

33.6

For x « 3 feet

[155QOO ^(3. 89+1. 0279-2x. 9124)3 (3. 89-3)

vx

22.8
For x - 3.73(total recoil)

Velocity » 0

_^_^^^___^____

Calculation of Guide and Packing Frictions.
g

2oKdb
Guide friction Rg - - - - approx.

u - .15

K = 155000

dfc * 15 .5 '(in) distance from center

of gravity to resultant pull.
1 = 37 + 48=85"(in) mean distance

between clip reaction.

2*. 15x154725x15. 5

.-. Guide friction - — - — — = 8450 Ibs

85

Stuffing Box Friction

Recuperator stuffing box
Diani. * 2.169

Bear sleeve - .5"+. 875"

contact

Inner packing ring - .787
Gland - .87

544

Recoil stuffing box
diam. = 4.728

Rear sleeve - .75*. 5
> Inner packing ring .787
Inner gland - .866

(Spring pressure + 0.1 pressure)(.75 diam. x H
.09 x length of contact) Formula.

1058
Spring pressure from drawing

10.124

.785 (6.4375«-5 .3437* )
104 Ibs/sq.in.

Oil pressure in recuperator = 576 + 1350

A . Initial Final

- - - = 963 Ibs/Sq.in

I

Oil pressure in recoil

2222+1670

= - » 1946 Ibs/sq.in.
2

Recuperator stuffing box diam. = 2.169 length of
contact (dermatine )=.787

Friction - . 75x2 .169x3 . 14" (963 +104)x.09x. 787»375.
Recoil stuffing box diam. =4.728 Length of contact

« .787

Friction « . 73x4. 728*3. 14* (1946+104)* .09x .787=1572
Total stuffing box friction = Recoil stuffing box

friction + Recuperator
stuffing box friction

» 1572+375*1947 Ibs. Total stuffing box friction.
Total friction » guide * stuffing box.

- 8450
_____ * 1947

10397 Ibs.
Calculation of Throttling Areas .

545

2[K- -(b+X-2Xr)](b-x)

«
C A

13.2 /K-pa-rRt+Wr sin 0

But

'[K- 7(b+X-2Xr)J(b-x)

•» V,

C A* Vx
W,

13.2/K-pa-Rt+Wr sin 0

Pa = Pai x Ar^PP1"0*)3*11^*4! pressure x ef-

fective area of piston
= 576 x 35.766 = 20600
C = 1.39 (constant)
A = 35.766

Hr sin 0 = 15790 x .0848 = 15550
Rt=guide friction + stuffing box friction =

10,000 Its.

K = 155000

Vx= take the values as calculated for vol. curve,
From calculations:

when x = 3.57"

V = 36.449ft.sec.

wx= .061 x 36.449 = 2.223 x 2 = 4.446
When x » 10.39in.

V = 42.118ft.sec.

wx= .061 x 42.118 - 2.5691 x 2 * 5.1382
When x = 10.94in.

V = 41.7ftsec.

*rx*.061 x 41.7 = 2.5437 x 2 = 5.0874

When x * 1.5ft.or 18in.

Vx = 37.4ftsec. 3

W L39 x 3S.7662

x 13.2 /154725 - 20600-10397+15550

.061x37.4=2.2814 sq.in.*2 rods=4.5628

546

l&

^

<s

s
fe

I

X,

Y

D/>

t

•* 5-

3&W6?

§ *-

_

547

546

When x
V,

2ft.or 24in.
•33.6
« .061 x 33.6

2.05 sq.in. x 2 = 4.10

When x - 3ft.or 361 n.

Vx- 22.8

wx». 061x22. 8-1. 28 sq.in.x2«2.76
When x « 3.75ftor 44.8in£total recoil)

w - 0

Comparison of Throttling Areas.

I no b a a Recoil

Calculated Area of

Orifice (2 rod.)

Frenoh Value

3.5-7

4.«46

4. 413

10.39

9. 13**

b. 129

10.94

5.08I74

5.084

18.

4.5628

4.54

24.

4. 10

4.08

36.

2.76

2. 69

44.8

0.

0.

SCHNEIDER COUNTBR RBCOIL.

The counter recoil is divided into three
periods:

(1) The accelerating period, the

: counter regulation being controlled
by a constant orifice through the
buffer in the recoil rod.

(2) The retardation period, the count-
er recoil regulation being controlled
by a variable throttling orifice
through the buffer head.

549

(3) A constant orifice period at the
end of recoil, the throttling orifice
being very small and the displacement
a very small part of the recoil.
The displacements corresponding to (1), (2)
and (3) are 1Q, lb and 1Q respectively.

Counter Recoil Data.

Length of constant orifice 1Q = 31.3 inches

Length of variable orifice 1^ = 7.85 inches
Length of constant orifice

at end of recoil 1 = 5.68 inches

b = Total c 'recoil 44.83 inches

Constant orifice period 0.7 b
Variable orifice period 0.175 b
Constant orifice at end of

c 'recoil 0.125 b

where b » length of recoil

There being 2 recoil brakes, we have for the
buffer reaction:

. «»•*;»;

B- *

where A^ = area of one buffer = 9.859
aQ » area of constant buffer

orifice = .0664 sq.in.

ao = area of constant

buffer orifice at end
of recoil - .022 sq.in.

Considering the c 'recoil at horizontal elevation,
during the constant orifice period, we have

•o »o 2-3026

where A = load on air - friction
= Py - I R

550

and C0 «

ZR

2c Ab

^•^•MM

175

6290 Ibs.
490

2.78 x 940
175

15

A x

Total

L OT*

2o(a«-*M

V

Buffer

in.

* in.

»*

2.3 »»*

force
t

. A

. 4

397oo 3415V*

. 198

1.16

4550.

. 8

1. 2

37860

.386

2.67

243OO.

. 8

2.

36860

.386

3.05

31700.

1.

3.

35810

.493

3.3

37100.

1.

4.

34910

.493

3.23

35500.

1.

5.

33910

. 493

3.18

34400.

1.

6.

33160

.493

3. 13

33500.

1.

7.

32510

.493

3.09

32900.

5.

12.

28710

2.5

2.91

29000.

5.

17.

25710

2.5

2.745

25720.

5.

22.

22710

2.5

2.5

22700.

5*

27.

20710

2.5

2.46

20750.

4.3

31.3

18910

2.13

2.36

18950.

Beginning of Variable Orifice.
After this period the unbalanced force was
assumed constant.

* ib * r M*o - *;>

.3-. 25

0 » 245 (-

.655

-) = 1880. # = unbalanced force

175 **

551

I n o b • •
Reooil.

x in

1880 x

1300-1800

x
V Total lb«.
Buf f »r

245

32.3

33.3

.0833
. 1666

157.
360.

4.61?

3.94

2. 16

1.99

20500

20200

34.3

35*3
36.3
37.3
39. 15

.25
• 333

.416

• 5
.656

47o
627.

780.
940.
1230.

3-38

4.75
2. 12
1.47
.286

1. 84

1.66

1.45

1. 215
.52

9900
19500
190OO
18500
18000

40.

.5

I750u

17000

42.

44. 83

• 5
.0

16500

16000

3 o i n t X Foot

45 80 z

323.5- 4580x
^215

V.I

Le » d

X

458OX

245

f.«.

on

Air

1

.0833

382

2833

11.

5

3.

39

28400

116

2

.1666

762

2453

10.

3.

16

27600

108

3

.25

1145

2070

8.

45

2.

91

26800

. 102

4

• 333

1525

1690

6.

9

2.

63

26000

.092

5

. 416

1900

1315

5.

38

2.

32

24600

.083

6

.5

2290

925

3.

78

1.

945

24400

.065

7

.656

3000

215

8.

76

*

94

21000

.044

ftHKttlDgB

X • any interval

0 • unbalanced force

Mr = mass recoiling parts

Vo * max. velocity of o 'recoil

Vx » velocity at any point

Vx * velocity at beginning of period le

1 > length of constant orifice period in feet

552

1^ s length of variable orifice period in feet
lc = length of final period for c 'recoil in

feet

Pa = load on air in Ibs.
R » Total friction

2K*AV

= total "buffer force

175 W»

b = length of c'recoil (ft)

Period 1Q
2K2A»Va

Pa - R - = 0 acceleration » 0

175A»

Assume velocity of 3.5 ft. per second and solve for
orifice W,

Period lc

pa - R = 0

175 W*

Assume velocity of 1 ft. per sec. and solve for
*x

Period lu

x =

Knowing V solve for Kx for various points

553

175

Pa = R Solve for Wx

ST. CHAMOND RECOIL.

ST. CHAMOND RECOIL The type of St. diamond brake
SYSTEM. here discussed, consists of three

cylinders; a hydraulic brake
cylinder, a recuperator cylinder
containing the floating piston

which separates the air and oil, together with a
regulator valve for throttling the oil between the
hydraulic and recuperator cylinders, a third cylinder
serving as a part of the air reservoir and therefore
communicating with the recuperator cylinder air
volume, the remainder of the third cylinder being
used for storing oil for the brake mechanism.

One of the peculiar features of this type is
the regulated spring valve where the main throttling
occurs. The valve functions somewhat as a pressure

regulator or governor, since if the pressure falls,
the spring reduces the valve opening tnereby in-
creasing the throttling drop and the pressure in
the hydraulic cylinder. The pressure in the recoil
cylinder, (i .e. the hydraulic pressure) is the sum of
the air pressure, plus ttie floating piston friction
drop, plus the throttling drop through the regulator
valve. At short recoil the air pressure is
necessarily small compared with the throttling drop.
The resistance to recoil is large and therefore the
recoil pressure large. This requires a large throttling
drop and the air pressure becomes necessarily small
compared with the throttling drop. The large
throttling drop requires a very small valve opening,
with a large pressure reaction against the valve.
To balance this reaction a very stiff spring is re-
quired. Such spring characteristics have been ad-

554

mirably met oy the use of Belleville washers. At
long recoil the resistance to recoil ia small, there-
fore the throttling drop is small, requiring a large
orifice area. Since the pressure in the recoil
cylinder is small together with a large orifice
opening, a weak spring with large deflection is
desirable. Such spring characteristics are best
met with an ordinary spiral spring. Hence, at long
recoil, low elevation, a spiral spring functions
alone, while at short recoil maximum elevation the belle-
ville and spiral spring function in parallel. The
regulator ia so designed that at low elevation only
the spiral spring functions.

To modulate or regulate the velocity of count-
er recoil to a low velocity, the pressure in the
recoil cylinder is lowered just sufficiently to

balance the total friction during counter recoil.
At the end of counter recoil the recoil cylinder
pressure is reduced to zero and the recoiling mass
is brought to rest by the total friction alone.
To reduce the pressure during the first part of
counter recoil throttling through a constant orifice
la effected in a separate passage way or channel
leading from the recuperator to the recoil cylinder.
At the end of counter recoil additional throttling
around a buffer rod and its chamber, is effected
reducing the pressure in the recoil cylinder to
zero or nearly so.

DESCRIPTION OF THE Referring to figure (10) is
OPERATION OF THE ST. shown a schematic diagram of
CHAMOND RECOIL. the operation of the St.

Cnamond recoil system for both
recoil and counter recoil.
Recoil:- During the recoil a flow or stream of oil

passes by the regulator valve from the hydraulic
to the recuperator (oil side)cylinder. The pres-
sure p of the oil against the recoil piston is re-

555

1

556

RECO/L REGULATOR

Fig- 6

557

duced by throttling through the regulator to a pres-
sure (pa) against tne oil side of the floating pis-
ton. Due to the friction of tbe floating piston the
air pressure pa is less than tbe pressure on the
oil side of the floating piston p^. The tension in
the recoil rod is balanced by tne total pressure on
the recoil piston plus the hydraulic piston friction
plus the stuffing box friction in the recoil cylinder.
The valve in the counter recoil orifice remains
closed during the recoil.

REGULATOR VALVE. The throttling during the recoil
is controlled by tbe regulator valve.
See figure (11). The regulator valve
consists of two parts: an upper stem
and the lower valve stew. The lower
valve stem is seated very carefully on a circular
seat at the top of the entrance channel. As the valve
lifts, the throttling area becomes the vertical cir-
cumferential area between the valve and its seat.
The spiral spring reacts on the lower valve stem.
The Belleville washers at the top of the upper stem,
react only on that valve stem. The upper stem rests
in a valve box or housing. To move the upper valve
stem (other than the slight deflection possibly

compressing the Bellevilles) the whole housing or
valve box is moved by a cam as shown in diagram.
The diameters of the upper part of tne lower valve
stem and the lower part of the upper stem, (that is
the diameter of the stems of the regulator valve,)
are tbe same. At short recoil the reaction of the
Belleville on the upper stem is transmitted by the
mutual reaction between the upper and lower stems
at their surface of contact.

The valve opening and consequent throttling
drop of pressure depends upon the deflection of the
spiral springs or Belleville washers, the spring
reaction balancing the hydraulic reaction on the
valve. Neglecting the small dynamic reaction, the

558

hydraulic reaction on the valve is fhe product of
the intensity of pressure in the recoil cylinder
and the base of the regulator valve, minus the
product of the intensity of pressure in the re-
cuperator cylinder and the effective area on the
upper part of the regulator valve. At long re-
coil, since the loner valve stem comes in con-
tact with the upper stem, the effective area on
the upper part of the valve is obviously equal to
the area at the base of the valve. Hence the
hydraulic reaction at long recoil is merely the
product of the difference in pressures between
the recoil and recuperator cylinders and the
area at the base of the valve. At short recoil the
upper stem of the regulator is brought down by the
cam at its top, until its lower surface is in con-
tact with the top surface of the lower valve stem.
The effective area, therefore, on the upper part
of the regulator valve equals the difference in
areas between the area at the base of the lower
valve stem and the area at the upper end of the
lower valve stem, or the area of the upper valve
stem; the two latter being always equal. Hence
the hydraulic reaction at short (or intermediate
recoil for the greater part of recoil) equals the
product of the recoil intensity of pressure and
the base of the valve, minus the product of the
recuperator intensity of pressure and the difference
in areas between the base and middle stem of the
valve, when upper and lower stems are in contact.
At long recoil the hydraulic reaction is balanced
above by the spiral spring reaction. At short
or intermediate recoil the hydraulic reaction is
balanced by the combined reaction of the Belleville
washers and the spiral spring though the latter is
negligible compared with the former.
COUNTER RECOIL. The regulator valve it
closed during counter recoil.
The oil flow during counter recoil, therefore, is

different from that in recoil. The valve is seated,

559

but the oil is allowed to pass through a very small
hole in its center. This orifice is constant through-
out the whole of counter recoil. There is another
channel for the oil leading from the bottom of the
buffer chamber in the regulator body. This oil
passes through a ball valve. As the floating pis-
ton returns to its initial position at the end of
counter recoil, the regulator rod enters the buffer
cavity, thus obstructing entrance of oil to this
cavity. This rod is tapered so that when it has
fully entered the cavity there is no clearance
between the rod and the entrance, and tne oil in
returning to the recoil cylinder nust all pass
through the central opening in the valve. By
neans of this regulation it is possible to allow
the gun to return to its "in battery" position
quickly, but its final movement is so controlled
that there is no ehock.

The throttling areas in the counter recoil
channels are so designed as to cause sufficient
throttling to lower the pressure in the recoil
cylinder that it nay practically balance the total
friction, during the counter recoil. At the end
of counter recoil this friction alone brings tne
recoiling mass to rest when it reaches the battery
position.

GENERAL THEORY OF THE Figure (11) shows the
ST. CHAMOND BRAKE. regulator valve stem for

both long and short re-
coil.
Let Rv » reaction on base of throttling valve

p * intensity of pressure ia recoil cylinder
pa * intensity of air pressure.

pa • intensity of pressure in recuperator

cylinder (i. «. on oil side of floating
piston)

a • entrance area of valve or effective area
at base of valve.

560

a = area of valve stem

Sb * spring constant of belleville washer
Ss = spring constant of spiral springs
C = effective circumference at base valve
h = lift of valve from initial opening
h » the initial compression of the spiral

3

valve spring at initial opening
hb * the initial compression of be Seville

washer at initial opening
w = throttling area
v = velocity of flow through entrance area

"a"

V =» velocity of recoil
A = effective area of recoil piston
d = density of oil

The hydraulic reaction, at long recoil becomes

i
Rv - paa, and at short recoil, we have, the value

Ry - pa(a-at). The belleville washer reaction,

oecomes Rb = Sb(hb+h) and the spiral spring reaction,

becoraes, Rs = Ss(hs+h). Hence at long recoil, we

nave Ry-paa = SS (hs + h) + F (1)

and at short recoil, we find

Kv-pa(a-at)*Ss(hs+h)+Sb(hb+h)+F (2)

that is, Rv-p^(a-at)=Sshs+Sbhb +h(Ss+Sb)+F

Now at intermediate recoil the upper valve stem
is separated from the lower valve stem by a distance
ho when the latter is just about to leave its seat.
If hQ is tne separation between the two stems before
recoil and if e = the initial lift of lower valve
stem required to clear the valve, then ho =hQ = e
Hence at intermediate recoil, we have
Rv-pa(a-at)=S8(hs+h)+Sb(n-n0+hb)+F, that is
Rv-pa(a-at)=Sshs*Sb(hb-h0)+h(Ss+Sb)+F (3)
Where F is the valve stem friction and will be
neglected, let

C0»Ssh8 and CQ =Sghs+Sbhb

co*ssns *3b(hb-ho>

561

The reaction against the base of throttling valve,
in terms of the pressure at the entrance to valve,

becomes, 2

Rv-p a = -^- (4)

g

where PJ = the pressure at a mid section in the

entrance channel of the valve.
Further 2

— * — = E + ht (5)
d 2g d

Neglecting the friction and accelerating head, ht
as snail we have, therefore

dv*
p = p - wnich gives the pressure in the

"& entrance channel in terms of the
recoil pressure (i.e. the pressure against the hy-
draulic piston) hence

dav dav dav
Rv-Pta+ -J— Pa 2g g

or

dav dv .

R=pa + = (p + )a (6)

2g 2g
Therefore at long recoil, we have

dv

(p+ )a=C +Ssh + paa

2g

and at short recoil we have,

dv*

(p+ )a=C0+(Ss+Sb)h+PaU-at) (8)

dv*
(p+ )a=C0+(Ss+Sb)h+pa(a~a ) (9)

2g

Considering now the main throttling through
the circumferential section, around the effective

562

circumference of the valve, we have, for the ef-
fective throttling area,

oh 1

w » — where K0 » ' Contraction factor of

Ko 0-775 orifice,

tbe corresponding pressure drop through the
valve becomes,

K!A"V'

p -

175(c«h«)

Further av » AV. hence V*=(T)* V*

hence lp+ •$£(£)

b * long recoil (10)

Sg+Sb recoil (11)

and
ll

f $-(-)* v* _c«_ if

S8+ Sjj mediate re-

coil (12)

Considering only the main throttling, or
ratber designing the recoil flow channels to have
throttling as compared with tbe throttling through
the regulator valve, we have

P * P * Pa hence we have the three fundamental
equations for the recoil pressure, in terms of the
velocity of recoil and tbe pressure in tbe re-
cuperator cylinder:

KSS8AV
p, - - - - +pa (13)

175Ca[(p+ -(-)" — ]a-C-P4a)' **

2g a 144 recoil

d A V*

175C*[(p+ — (-)* )a-C'-pa(a-at)a at

2g a 144 short

recoil.

AV

+ pa (15)

175C*[(p+ — (-)* — a-C0'-pa(a-a )]' _
2g a 144

mediate

recoil.

where the units are obviously, p'a and p in Ibs.
per sq.in.

V in ft. per sec.

Aiaiand at in sq. in.

d in Ibs. per cu.ft.

If further J0 * I » a~ * 144 then «9uati°n8 (13),

(14) and (15) reduce
to the simpler form

- :; - ; - z— (16) at long re-
175C [(p-pa)a+J0V -C0] coil

p_p „ - . - .. -
175 C [(p-Pa)a+Paa+J0V -CQ]

recoil
K*(Ss+Sb)A*V*

;; — (18) at in-
175C [(P-Pa)a+P;a+J0V -CQ"] ter.ediate

recoil

To compute p for any given displacement and cor-
responding recuperator pressure and recoil velocity,
we find the solution in the form of a cubic equation

The solution is as follows:
From equations (16), (17) and (18),

KV.AV '

175Caa»
p~pa "

564

or

P-Pa

VV*

175C8a2

lot

B , K A'V'S* or

175C«a* 175 C«a*

p-Pa * z

Jovico PaW'-Co

and or = a

a a

Then from the above equations, we have

or B=Z +2Z m+Zm

To eliminate the 2nd degree term, substitute,

2 a * 4 4

Z » X - r m hence Z -X - - • X + - n

O 39

and ,

3 3 9 •* 9 H *

Z — V^ O m Vo. » V

A G ifl A ~ Ql A"1" ~" T ID

Expanding, we find

4 » 8 s x ,,a 82..

B = X

3 2? 3

* 8- "' * ,8x - f .'

= X3 - - m*X - — m8

Further let N3 * — + -7 then X* - - m*X-N%3» 0
^7 m° 3

Solving by Cardan's method

^

— + / — — •» / — — / — — • ) m

2 4 730 2 4 730

565

2

X a Z

+ ~~ Hi

+ -

3

a 3

hence

*

N8

. 3 /

/i! _1_

2^

p »(/

2 *

4~ " 730 + 2~ " '

4 " 730

"3}

During the greater part of recoil except at the
very beginning and towards the Very end of re-
coil it has been found by actual calculations, that

the term —*•- becomes negligible in comparison with
730 ^ e

— and nay be omitted without appreciable
4

error.

The above equation, therefore reduces to the

simple form

2
p<= m(N - g)+ pa

Another and far simpler method for the com-
putation of p established by Mr. McVey, consists
in the construction of a table, with assumed values

of P ~ Pa.

The table is based on the two following equations
(neglecting dynamic bead as small)

(p-pa>A*PaAi s co+(ss+sb)h short recoil

(a)
(p-pa)A=C0+Ssh long recoil

pa A«Ya
and p-pl - 175C,h,

If ve assume a mean air pressure throughout, (the
error thus introduced having been found small),
we have

or =

ss

Assuming a series of values of (p-pa) «e obtain a
series of values for h, now from (b)

566

567

568

569

from which a series of values can be established
for corresponding values of p-paand h.

Knowing the retarded velocity for any given
point, the corresponding value of (p-pa) and h can
be picked from the table and knowing pa for the given
point in the recoil, the recoil pressure p is ob-
tained.

It is to be noticed, that substitution of (b )
in (a) gives a cubic with a second degree term, as
before. Thus no direct simple solution is possible.
The table method is recommended even for short re-
coil since the error introduced by assuming the air
constant is relatively snail.

GENERAL PROCEDURE FOR Due to the complexity of
CALCULATION OF RECOIL, the general equation of re-
coil no mathematical solution
is possible, except by ex-
panding into a series. Such

a solution of a recoil equation is known as the
"point by point" method and has been used before
in this text.

The object of actual computation of rec-oil
curves for a given type of mount is to ascertain
the ratio of the peak to the average resistance to
recoil at maximum and zero elevation. The average
resistance may be readily obtained in the preliminary
layout of a design and knowing the peak ratio for
a given type of mount, enables the peak resistance
to be obtained and the consequent stresses in the
carriage. Let

Vf0 = free recoil velocity at point "n" (i.e.
the velocity generated in the recoil-
ing mass by the powder pressure).
Vrn » corresponding retarded recoil velocity
Rn = total friction, stuffing box and guide

friction.

$ = angle of elevation of gun.
Now, the end pressures at the beginning and end of

570

<*JU-

recoil, becomes p-pi * — for

ft

a -at C0
p* Pa( - )* — for short recoil

A &

Since 0 = 0 at long recoil, we have for the re-
sistance to recoil, K » pnA +Rn for long recoil
K » pnA+Rn-Wrsin0 for short or intermediate recoil

Long recoil:

For 1st point long recoil,

Pao + + R

o

Vrt " vft

and knowing yri

3

, A* A* i A* A* i 2.

• a ( / — + / — — — — + / — — J — — — — — _\m+Dl

4 4 730 2 4 730 3 Pa,

m =

175C2a2 a

For 2nd point long recoil,

and knowing Vra

730

{>AtVr.S8 J0V«a -C0

B » V-- ; m = f —

175Caa» a

After a very few intervals the valve opens sufficient
ly so that the term

571

— rr may be omitted, then for point "n" at long re-

I o(j

coil,

(pmA+Rn)

Vn-V<vrrrvrm> ' At and knowing Vn

mr
2

Pn - m(N - -)+pa Obviously after the powder pres-
sure period Vfn-Vfm = 0 and we
have the simple dynamic equation of recoil.
Short Recoil:

The procedure for the calculation of the
velocity and pressure curves for short recoil is
exactly similar as- for long recoil.
For 1st point short recoil,

P« (— — ) + — +R -Wrsin0

c* 1 a a 1 *

»r, "f , -< ~ T5 > 4 *

and knowing V_
a * i

, /N8 /N«~ 1 /N A*~ 1 2.

a( / — + / — — + / — — J — — — — — )m+Dl

1 2 4 730 2 4 730 3' Pat

P
where

175C2aa a

for 2nd period short recoil,

p'A+R -W sin 0

V ~v =vf ~vf - C -
ii

and knowing Vr2 , p2 can be obtained by a solution
of the previous cubic equation. The greater
number of points of recoil excepting a few points
at the beginning and end may be solved with suf-
ficient accuracy by the expression,

2
pn=m(N- -gO+Pa "nere, as before,

572

V^-— K«A«V«(Ss+Sb)»

N - / — + — B »

27 •" 175C»a«

and

Calculation from constructed table of (p~Pa)» h,
and V~

The procedure here is exactly similar as above;
each preceding interval establishes a new retarded
velocity which from the table establishes a new
recoil pressure. This recoil pressure substituted
in the dynamic equation in turn establishes the re-
tarded velocity at the end of the interval under
consideration.

Judgment must be aged in the proper increments
of time to be used. The closer the intervals
the more accurate the velocity and pressure curves.
At the beginning and end, the time intervals should
oe taken smaller. During the major part of re-
coil the time intervals can be fairly large. As
a check during the powder period the retarded
velocity should be roughly 0.9 of the free velocity
of recoil.

CALCULATION OP THE VARIOUS In the calculation
FRICTION COMPONENTS DURING of the vertical pres-
RECOIL. sure and retarded

velocity curves for
the St. Chamond brake,

the frictions vary as a function of the pressure.
At long recoil the pressure variation is small
and we are not in great error in assuming constant
friction: with short recoil, however, a peak value
is obtained and with it a change in friction.

The frictional resistance opposing recoil are:
(1) Guide friction which is function
of the total pull.

573

(2) Stuffing box friction which is
a function of the recoil pres-
sure.

(3) Recoil piston friction which
also is a function of the recoil
pressure.

(1) The guide friction during recoil has been
previously expressed by the following equations:

2(pbe+Bb)+Wrcos0N
R = - n

* n

where pb is the powder reaction on the breech
c is the perpendicular distance between

the axis of the bore and a line through
the center of gravity of the recoiling
parts parallel to the axis of the bore.

B = pA, the hydraulic reaction of the re-
coil piston

n = coefficient of friction, from 0.15 to 0.
20

"b * distance down of the line of pull from
the center of gravity of recoiling parts

where XA and yt are the coordinates of the front

clip reaction and x^ and yg are the coordinates
of the rear clip reaction having axis and origin
through the center of gravity of the recoiling parts

Considering the somewhat inaccuracy of a
"point by point" method of computation, it is be-
lieved the following formula is sufficiently

accurate, 2nBb + nWrcos 0(xt-xa)
R = - -

C - 2nr
or when *~x *s small,

where r is the mean distance from the center of
gravity of the recoiling parts to fhe line of
action of the guide frictions.

574

(2) The packing friction formulas have been
already considered in more or less detail in
Chapter VIII. The stuffing box friction,

R8 » c^ + c; P

where

c

t

Ct- ndr(bf+af+atft)
(3) The hydraulic piston friction,

Rp=C" * C

ndp0(bf+afi)

From the above formulae

p * recoil pressure in Ibs. per sq.in.
P0=intensity of pressure caused by Bellevilles

or packing springs in Ibs. per sq.in
Rb * belleville or packing spring reaction on

annular area of packing spring at as-

sembled load in Ibs.
dr * diam. of piston rod in inches.
do = outer diam. of stuffing box packing

ring in inches.

d * diam. of recoil cylinder in inches.
di=inner diam. of piston packing ring in

inches.
b = width of leather contact of packing in

inches.
f * corresponding coefficient of friction *

- - silver contact of flap of one flange of

packing ring in inches.

ft» coefficient of silver friction » .09
Then po becomes,

for (Dp

575

In summing up the component frictions, we
have

nWrcos 0(xt-xf)

C-2nr

RS = c; + ct'P

Rp - ct" + c; P

hence R - (C»*C|[*Kt )+(CJ + C;1 +K2) p

» c4+csp

showing the total friction resisting recoil is a
linear function of the pressure in the recoil
cylinder.
Floating piston:

The oil pressure in the recuperator cylinder
during the recoil is greater than the air pres-
sure by the drop of pressure caused by the float-
ing piston friction. In the previous recoil
equations, the recuperator oil pressure has been
used in place of the air pressure. To compute
this pressure knowing the air pressure, it is only
necessary to compute the floating piston friction
drop. In the discussion of the floating piston in
Chapter VIII, we have RfsCt+Capa where
Ct- Kdl(bf+aft)(p0+p0)+20tflPq]
Ca= nd[2(bf+aft)+2at£a

For symbols see discussion of floating piston
in Chapter VIII. All dimensions may be expressed
in inches and pressures in Ibs. per sq.in. in place
of the center of gravity system as used previously.
The resulting friction Rf is therefore in Ibs.

The drop due to friction, becomes,

Rf
pa - pa » - where Aa is the area of the floating

Aa piston or recuperator cylinder in sq,
inches .

576

577

GENERAL THEORY OF Counter recoil is divided in-
COUNTER RECOIL. to two periods, (1) the first

period or constant orifice period
and (2) the second period or
buffer period wnere the main re-
tardation takes place. The second period is the
critical period in the design of a counter recoil
system, since with field carriages the stabilizing
force of counter recoil is relatively small, there-
fore too rapid retardation of the recoiling mass
will cause the mount to be unstable on counter re-

coil. Let

•

A = effective area of recoil piston

Ko= contraction factor of constant orifice

Kt* contraction factor for variable orifice.

WQ= area of constant orifice

«x= variable area of buffer throttling
R = total friction

pa= recuperator oil pressure
Pa * Pa^ equivalent recuperator pressure on
recoil piston

C0* • = throttling drop constant for con-
176 stant orifice.

K*A*
Cx= - * throttling drop constant for buffer

175 orifice.

Ro=l?rsin0 + R a resistance constant.

For 1st Period of Counter Recoil:

Considering the notion of the recoiling mass,
from tne initial displacement of out of battery,

we have dv

pA -Wrsin0-R=mrv —

but

P '•

Assuming the throttling drop is entirely
through the constant orifice during the first period

578

„« .3 2

KQA V dv

of recoil. Hence p^A - Wrsin 0-R=rorv —

175w* dx

or p v*

•

Since pa is a function of xtthe equation is
not possible to integrate directly, but by divid-
ing the constant orifice period into several in-
crements, and taking a constant air pressure equal
to the mean air pressure for the interval, we get
a very close approximation of the true velocity of
counter recoil by the following solution, 9

ror » dv
dx =

Integrating, we have

mrwo CjjV* Cov

Substituting for the base 10, we have

Cov* Crtv*

logf(pa-R0) -]=logUpa-R0)'

as 2 ivz V>'4miuz

"o o ^.omrwo

From this equation, knowing the velocity at
the beginning of any arbitrary interval and with
the mean recuperator pressure we can obtain the
velocity at the end of the interval. It will be
found that fairly large intervals may be assured
with considerable accuracy, providing the air
pressure does not vary greatly.

The velocity curve for the first period should
be continued from the out of battery position to
x = b-d, where d= the length of the counter re-
coil corresponding to the buffer length.

Let vj, = velocity of counter recoil at entrance
to buffer.

For 2nd Period of Counter Recoil.

The recoil displacement is d, and the initial

579

velocity v^ . In order to be assured that the
c 'recoil is completely checked, the counter re-
coil energy of the recoiling mass at entrance to
buffer should be dissipated in a distance somewhat
less than d, from 0.7 to 0.9d, depending upon the
design constant of the recoil system and gun. Let
k * the proportional distance of d that the recoil
energy is to be dissipated along.

For counter recoil stability the minimum force
during the buffer period is obviously obtained by
using a constant force during the entire period.

There are two methods consistent with counter
recoil stability:

(1) When the total friction is small
compared with the overturning force
permissible with counter recoil
stability, by "bringing the recoiling
mass to rest into battery with

t"he friction alone.

(2) Khen the total friction is greater
than the overturning force permissible
with counter recoil stability, by
bringing the recoiling mass to rest
into battery by a force equal to

the total friction minus a recoil
pressure exerted on the recoil
piston.

In method (1) obviously, for a given kinetic
energy of the recoiling mass at entrance into
buffer, the recoil displacement during the buffer
action is fixed.
Method (1)

We have for the required recoil displacement
during the buffer action:

1 2

-m_vb

where k = 0.7 to 0.9

kR

where R is the total recoil friction (guide,
stuffing box and piston friction).

580

The length of the buffer in the recuperator
cylinder becomes,

d' - — d
•I

The velocity curve is evidently a parabolic
curve against displacement, that is

dv

R * - m_ v —
dx

x v

/ Rdx = - mr / v dv
b-d vb

R(x-b+d) = -£(v - v«)

hence

/ 2R
v / v* -- (x-b+d)

m

Since it is assumed that p = 0, we have

substituting for v, we have wx in terms of the dis-
placement x from the out of battery position,

vg - — U-b+d)
"x • M "o "

-- (x-b+d)]

where b = length of recoil in ft. and d =
length of buffer recoil in ft.

x - counter recoil displacement from out of

battery position in ft.
The throttling drop

581

— — through the constant orifice has been found

1 7 i M

° by calculation to be small as compared with
the throttling drop due to the buffer. Therefore
a simplification in the calculation may oe made by
omitting this term, hence

. —7- and substituting for v, we have
13 • 2

KtA /mrvb-2R(x-b+a)
HX m / approx . which gives

13.2 mrp^ the required throttling
area in terms of the displacement of counter recoil,
(x is measured from out of battery position in ft;
b - recoil displacement in ft. and d = recoil
buffer displacement in ft).
Method 2

If h = height of center of gravity of recoil-
ing parts above ground (practically from ground to
axis of bore) and wg = weight of entire carriage
and gun and C's - distance of ws from wheel contact,
then critical stability (at 0° elevation) we have,
(R-pA)h=Nsl^ using a factor of 0.8, we have

W 1 '
s s

R - p A * 0.8 where R is the total friction,

h

but now a function of the re-
coil pressure, let R = Ct+Cap then

Wsl'

C +P(C -A)=0.8 — —
1 h

W 1 '
or _ wsxs

0.8 — C h

b l

p = where C - guide friction as-

°(Ct ~ A) sumed independent of p + that

part of packing friction in-
dependent of p.

C * that part of packing friction dependent upon
p. The counter recoil velocity curve, "becomes
during the buffer recoil,

582

v£ - (x-b+d) and for the pres-

n\r

sure p in the recoil

cylinder, we have

K

Pa -

175w«

» P

hence

KIA v/""

1

wx - /

13.2

K*A2v2

(pa-p>-

or in terms of
the displace-
ment x,

(R-pA) (x-b+d)

2(R-p A)

175(pa-p)w*-K§A* [vg (x-b+d)]

mr

Neglecting the constant orifice throttling drop,
we have the following approximate formula

12.2

It should be carefully noted that if v^
is allowed to become too great it may be found
very difficult to prevent the final check of
counter recoil without shock sure with even
prompt throttling by the regulator the kinetic
energy ot the gun may overcome the the opposing
friction and cause bumping.

DESIGN FORMULAE ST. In the preliminary design
CHAMOND RECOIL SYSTEM, of a St. Chamond recoil sys-
tem, we must consider the
following.
(1) The proper weight of recoiling

mass wit"h given ballistics and allow-
able recoil at maximum elevation
for minimum weight of the total
mount, gun and carriage.

583

(2) Tha length of recoil at zero ele-
vation consistent with stability.

(3) The total resistance to recoil
at short recoil maximum elevation
and at long recoil, zero elevation.

(4) An estimation of the guide frict-
ion and packing frictions for both
recoil and counter recoil.

(5) The recuperator reaction re-
quired to hold the gun in battery at
Maximum elevation.

(6) Limitations of recuperator
dimensions .

(7) The calculation of initial air
pressure and air volume, final air
pressure and air volume. From this
the equivalent air column length.

(8) The calculation of strength of
cylinders and proper thickness
between walls.

(9) The layout of the recuperator
forging distance of center lines of
cylinders with respect to axis of
bore, location of trunnions, etc.

(10) The calculation for maximum and
minimum throttling areas.

(11) The calculation for entrance chan-
nel area to regulator valve, regulator
dimensions and channel areas around
and leading from the regulator orifice.

(12) The reactions on regulator valve
corresponding to deflections at maximum
and minimum opening and the design of
spiral springs and belleville washers.

(13) The design of cam mechanism for
changing the initial opening to
regulator valve for decreasing the
recoil on elevation.

(14) The design of packing for float-

584

ing piston, recoil piston and

stuffing box.

(15) The design of the counter re-
coil and chamber, throttling grooves
and constant orifice with its chan-
nel leading from the inside end of
the buffer to the recoil cylinder

(16) The layout of gauge and pump
details and all other details.

(1 ) Proper weight of recoiling mass:

From "General design Limitations" we have

»_ * / kk1 where

"c

k T

and

., _ gUv + m 4700.)'
2b

m = mass of projectile
I * mass of charge
b = length of short recoil in feet
R » recoil reaction in Ibs.
g = acceleration due to gravity (ft/sec)
• wc= weight of carriage excluding recoiling

mass in Ibs.

k may be obtained from table in Chapter VII
or the ratio

wc

— may be computed from a similar well

designed type of carriage, using

a somewhat lower value of "b" according to the
.judgment of the designer in improving the weight

efficiency of the mount proper over a similar
previous design.

(2) Length of recoil at zero elevation,

Prom pressure curves obtained experimentally
it was found that the resistance to recoil at zero
elevation is practically constant throughout the
recoil.

585

Let b = length of horizontal recoil consistent with
stability in feet

wv+w4700
Vf = - = free velocity of recoil.

w = weight of projectile in Ibs.
w * weight of charge in Ibs.

Wf = weight of recoiling parts in Ibs.

v = muzzle velocity (ft/sec)

Vr« 0.9 Vf(approx.) = velocity of restrained

recoil.

u * travel of projectile up bore in feet
A » recoil constrained energy = 7Jtirvr
E * recoil displacement during powder period

= 9 06.

* £.64

"r

Overturning moment
C * constant of stability = - — . . , .

Stability moment

Wg * weight of total mount, gun and carriage
lg * moment arm of wg about spadepoint.

d = perpendicular distance from spade point to
line of action of the total resistance to
recoil.

Usually £5 = 0, cos 0 = 1 and d = h = height of

axis of bore above ground.

then

Wrcos

and when 0 * 0, we have,

1 / 2AW_h
b - E + — [Wsls ± /Ofjjlp8 ]

Tir P

•V «» *^

Ordinarily the constant of stability will be as-
sumed at C * 0.85.

For rough estimates, especially wnere the
length of recoil is comparatively long,

Cs Vrcos 0

where A
2A

586

1 -7- " r mrv!

(3) Resistance to recoil, short and long recoil,

For design calculations, Bethel's formula for
the total resistance to recoil is sufficiently
accurate. Let w = weight of projectile, Ibs.

Vfr = weight of recoiling mass, Ibs.

M = travel upbore in inches

d = diam. of bore, inches.

v = muzzle velocity of projectile (ft/sec)

Vj = max. free velocity of recoil (ft/sec)

bs = length of short recoil at max. elevation
in ft.

bj, = length of long recoil at zero elevation
in ft.

w = weight of powder charge
Now for the free velocity recoil, we have

w 4700 + wv
v = -

then at maximum elevation and short recoil, we
have

Ks'1.05[

8 u + (. 096*. 0003d )m —
Ds v

(where 1.05 accounts for the peak effect due to
throttling) and at horizontal elevation, long re-
coil, we have

y — the peak effect
2g hh+(.096 +.0003d)m-£ being zero

(4) Guid*e and packing frictions - Recoil and
Counter Recoil:-

In the calculation of guide friction during
the recoil consideration must be given to the pinch-
ing action at the clips due to the pull being

587

usually below the center of gravity of recoiling
parts. Failure to consider this fact will give
erroneous results for the friction at high
elevation. Also the recuperator must be de-
signed not only for the weight component at max.
elevation but the friction just out of battery.
Since at the end of counter recoil we have the
full air pressure acting on the recoil piston,
the pinching action and corresponding guide frict-

ion being a factor of importance. Let

B = the total braking including the

packing friction of recoil piston and
stuffing "box, Its.
d^ = distance down from center of gravity

of recoiling mass of line of action of B
in inches.

pA = the recoil reaction, Ibs.
n = coefficient of guide friction
x^ and xa = coordinates in direction of bore
of clip reactions measures res-
pectively from center of gravity
of recoiling parts in inches.
1 = x + x = distance between normal clip

reaction, in inches.
_ Rfi = total guide friction.

From a similar previous design, a value of b and 1
may be assumed.

Prom ChapterlV, we have

2nBdb+nWrcos

and for a first
approximation

assume x

then

R =

2n8ab

but K = B+Rg- Wr sin
(K+Wrsin 0)1

1+2 n

588

Knowing K and assuming 1 and b, we have
BspA+Rs+Rp where Rg - stuffing box friction

R = recoil piston friction.

Since the design and estimation of packing
friction is in greater part based on previous
empirical data, the width of packing and cor-
responding dimensions being entirely an empirical
matter, we must estimate the proper value from
data on previous satisfactory packing used on other
guns.

Now in general the packing friction may al-
ways be represented by the following equationt-
R » CY+C p. where Ct is the friction component
caused by the springs or Belleville washers. Since
the object of the Believilles is to compensate for
the deficiency of the oil or air pressure normal
to the cylinder due to Poisson's ratio, if we
know tne maximum pressure and assume dimensions for
the packing we may compute Ct as well as C? and
thus estimate the friction at any other pressure.

Maximum pressures normal to cylinder should be

taken as follows:

Pn

k

Hydraulic piston

°-88 Pmax.

0.88

Stuffing box

0-86 pmax.

0.86

Floating piston:

Air side

i-20 Pa max.

1.20

Oil side

1-38 Pa max.

1.38

Knowing the maximum pressure normal to the
cylinder "pn" we have,

max.

n dC.05 b+.09(a + T^Pn approx.

0

589

s s

1 I

tl

590

where d * diam. of piston rod or cylinder in inches,
b * width of leather contact of packing
a = total depth of one silver flange of pack-

ing cup in inches.
at = total depth of outer silver flange.

Prom the above equation, we have

Ci*c2 Pmax. = nd[.05 b+.09(a+

pn where

Pmax.

tne maxiraum fluid pressure, Further,

K d 0.731 .05b + . 09 (a+ r^) )proax % hence

o

a

C » K d[ .05b+.09(a+ — )](pn- 0.73 Pmax)

a

C = n d 0.73[.05b+.09(a+ — )]
8 2

As a guide for suitable values of a, at and b
with given maximum fluid pressures, the following
table has been constructed of values used in cer-
tain experimental recuperators.

75 m/m Model of 1916 MI.

Pmax

Ibs/sq.

in.

Recoil piston
Stuffing box
Floating piston

0.14" 0.18" 0.19" 5120
0.14" 0.18" 0.19" 5120
0.09" 0.18" 0.29" 1270

3.3" Model of 1919.

a a

b Pmax lbs/sq

in.

Recoil piston 0.137"

0.233" 5500

Stuffing box 0.137"

0.233" 5500

Floating piston 0.137"

0.194" 1850

591

4.7" Model of 1906

Pmax
Ibs/sq.in,

Recoil piston 0.128" 0.128" 0.335" 3820

Stuffing box 0.128" 0.128" 0.335" 3820

Floating piston 0.128" 0.200" 0.335" 1200

4.7" Model of 1918.

a at b Pmax

Ibs/sq.in,

Recoil piston 0.156" 0.218" 0.218" 4500

Stuffing box 0.156" 0.218" 0.218" 4500

Floating piston 0.125" 0.218" 0.35" 2300

The dimensions a, ai and especially b increase
somewhat with the diameters of the cylinders (that
is with the caliber of gun) as well as with the fluid
pressure. Let C^ and C^ be the packing friction
constants for the stuffing box.

C" and C" be the packing friction constants
for the recoil piston

4- U n to D ^R ^°~ V *•* ^°O / \ V • W / P

tnen Kg v p i t x a

= Co +Q p
i °2P

Therefore, the recoil reaction, becomes, for any
pressure pt

(K+wrsin0)l

pA = (C0 +C p)

1+W n db

If we assume the maximum recoil pressure pmax
corresponding to maximum elevation #roax we have

C.

d[.05b+.09(a+

where pn = k pmax, k being obtained from the
previous table. Therefore the effective recoil
piston area, becomes

(K+»rsin0_ax)l a

k n d[-05

592

- .

(1*2 n db)pmax

In general pmax = 4500 Ibs. per sq.in. but as
we shall see in 6, with guns of low elevation and
with reasonable horizontal stability, the max.
pressure may be necessarily smaller than the pack-
ing limit pressure of 4500 Ibs. per sq. inch. The
assumed max. pressure for calculation of packing
friction and in (5) the recuperator reaction is at
this stage a questicn of experience.

The guide friction when the gun is in battery

becomes, 2n Brdb

R =

l+2nr

where Bv = paA-(R-s + Rp) i.e. the tension of the rod

in battery

r = distance down to mean friction line
1 = distance between clip reaction in battery.
n = coefficient of friction = 0.15 to 0.2
b = distance from center of gravity of re-

coiling parts to line of action of By
Further the value of Rg^R in battery becomes,

To compute the drop of pressure across the float-
ing piston friction, we have

Rf *+C"'+C£ ' — — — approx.
while pa~Pa * —
hence 2C"' + (C

Pa = — h — :

, _ (2Aa-Cg)pa-2Ct
C +2A

.

which gives the air pressure in terms of the re-
cuperator pressure (for recoil computations) or

593

the recuperator pressure in terms of the air pres-
sure in terms of the air pressure (for counter re-
coil computations).

In a preliminary layout, we are not greatly •
in error in assuming either

Pa-Pa =

approx. drop due
or

to floating piston.

"a

(5) The recuperator reaction required to hold the
gun in battery

To ensure a sufficient margin for the holding
of the gun in battery and overcoming the friction
in battery, an excess of 20* to 30* is used over
the minimum recoil reaction, hence

2nKydb

_ Rg = - ; k = 1.2 to 1.3,n»0.15
1+2 nr

Kv
and B8+Rp=C,+ C, —

where Ct=0. 15* (dr+d) (dp+d)[.05b + .09(a+ T

a

C =0.73n(dr+d)l .05b

t

and for a trial value, Pmax = 4500 Ibs. per sq.in

ps

A = -
4500

d * diam. of recoil piston

dr=diam. of piston rod

dfc38 distance down from center of gravity of re-

coiling parts to line of action of Kv
Substituting in (1), we have,

2nk Kvdb ITV

~ - +Ct+C2 ~
1+2 nr A

594

kWrsmn0_ax+C

bence K = - - - Z£* — i - (2)

2nk d C

l+2nr A

(6) Limitations of Recuperator Dimensions:

In the design of a recuperator layout, we
must consider the proper ratio of area of re-
cuperator cylinder to effective area of piston,
the limitations of areas based on this ratio and
on the maximum allowable packing pressure in the
recoil cylinder as well as on the difference
between the horizontal pull and recuperator re-
action at maximum elevation. If now,

wn= max. area of orifice at horizontal re-
coil in sq . in.

A = effective area of recoil piston in sq.in.

Aa * area of recuperator cylinder or floating
piston in sq. in.

V = max. recoil velocity in ft. per sec.

Pn= horizontal pull in Ibs.

ps= pull at maximum elevation in Ibs.

Kv= recuperator reaction at maximum elevation
in Ibs.

1

K = - = reciprocal of orifice contraction
0.773 constant.

wc= channel or port area at cross section beyond

regulator valve in recuperator cylinder
in sq. in.

Ws* area of channel section through diameter

of regulator valve in sq. in.
a = entrance area to regulator valve.
d = diam. of regulator valve.

r = ratio

Floating Piston Area

of

Effective Area of Recoil Piston

Now the maximum throttling area w^ i» the
limiting throttling area, since all constant port

595

or channel areas in the recuperator should bear given
ratios with respect to this area and must be suf-
ficiently large as compared with wh so that there
is no appreciable throttling through them, or loss
of head due to friction or acceleration.

The following table gives ratios of channel
or port areas in the recuperator with respect to
the .maximum throttling area w^ and the area of
the recuperator cylinder or floating piston.

Model

"h

"c

we
C =»
1 a,

wh

*c

C*T

*a

-.
*>*-.

4. V "-M. 19O6

0. 38O

1.61

4. 235

0. 126

0. 608

4.7"M. 1918

0. 854

3.6-7

4. 3OO

O. 207

O. 64O

3. 3"-M. 1918

0.267

1. 11

4. 160

0. 186

0. 470

75»»-M. 1916

0. 1*75

o. 7s

4. 47O

O. 120

0.371

4. 3OO

o. 160

From the above table, the following design
constants will be used based on satisfactory layouts:

C = — = 4.3

1 wh
Now

= f = 0.16
C.

c C. ,

*•' a • -0373 r "

Considering the throttling through the regulator
orifice, we have

2

»h

KVv*

(1)

175(pn-Kv)

and substituting for wh=.0373 r A, we find
4.11 AV2

that is r = 2.62 V

Pb'Ki

(2)

Hence, the ratio of the recuperator area to ef-
fective area of recoil piston varies as the square
root of the effective area of the recoil piston,

596

and for minimum recuperator area we must have min-

imum effective area of recoil piston.

Hence the recuperator area always varies as the -
power of the effective area of the recoil

piston.

Now for minimum neight of the recuperator
forging it is important that the cylinder areas
be made as small as possible. Therefore the re-
coil piston area in general is limited by the
maximum allowable pressure in the recoil cylinder
consistent with the packing pressure limitations.

If the packing limiting pressure is taken at
4500 Ibs. per sq. in., then

A = -£i-

4500
Substituting for A, we have

.039V / - ^«
Pb~Kv

How r is limited by the following considerations:
(a) In an ordinary layout, the
initial volume of the air, may
be represented as the sum of
_, air column in the recuperator

cylinder plus the air column in
the air cylinder, that is

where kt = initial air column
length in re-

cuperator length of

recoil
and k ~ air column length in air

cylinder length of recoil
k may be obtained from the fol-

597

lowing table:

b

X 1

b

b'..x

0 »«x

b '

k» b

wax

4. 7"-M. 1918

40

28. 27

1. 41§

o. 7o;;

4. 7"-M. 1906

7o

56.50

1. 240

0. 807

75»m-¥. 1916

46

43*08

1. 07O

0.937

3. 3"-M. 1918

60

47.67

1.260

o.795

1. 246

0.811

Therefore «e may assume Ki=0.8 and kf will be
assumed ka= 0.7, hence kt + ks = 1.5

The initial volume as shown in (7) may be
represented in the recoil piston displacement and

the ratio of the final and initial air pressures.

Paf
If o = and k = (1 to 1.41 use 1.3)

then

when

m = 1.5

(6)

= 3.73

.k - 1

2.0

3.0

= 2.42

- 1

1.75

- 1

Using a ratio of m = 1.5, and combining Eq. 4
and 5, we have 3.73 Ab=1.5Aab hence

= r = 2.5

598

Paf
If a lower value of m = - is used, then r > 2.5

Pai
where as with a higher ratio, r < 2.5

If we continue increasing m for the minimum
permissible value of r we soon arrive at the
limitation where the maximum possible recoil of

the floating piston limits the ratio r. since Aa

amin

bmax - A b- Then amin b

max

On the other hand to obtain a value of rmjn=
1.25, would require a high value of m, approx. a
3.0, and the temperature rise of the air would be
excessive and very injurious to the floating pis-
ton packing.

Further, at horizontal recoil, where a stability
slope for the total resistance to recoil is highly
desirable, we have the minimum throttling drop
due the small value of the pressure in the recoil
cylinder at horizontal recoil. The peak effect of
the throttling drop is thereby reduced, and since
the pressure in the recoil cylinder is the
throttling drop plus the recuperator pressure, a
large increase in the final air pressure over the
initial will overbalance the decrease in the

throttling drop towards the end of recoil. There-

fore, a large value of m = ¥-&£ must result

in a rise of the total pai

resistance to recoil towards the end of recoil

which is entirely inconsistent with the re-

quirements for horizontal stability

If, therefore, at horizontal recoil, we

limit the ratio m =s to that value which gives

us a constant

resistance throughout the recoil, we have

neglecting the slight throttling drop at the end

599

of horizontal recoil. Substituting for
wh».0373 r A K

0.773

and

we have, 2

0.145(m-l)r2=6.87 - (8)
Pai

From the following table, knowing V and
pa^ and from the above equation (m-l)ra, we can
readily obtain r or corresponding value of m.

3 CYLINDERS

2 CYLINDERS

r

r2

r2

6. 88

(m-1)

V2
Pai

m-1

V*
*ai

1.5

2. "75

.3273

1. 147

.3787

1.547

.5063

1.6

2. §6

.3*724

1.015

.3780

1. 345

. 5001

1.7

2. 89

. 4204

. 910

. 3824

1. 188

.4994

1. 8

3.24

. 4714

. 824

. 3884

1.064

. 5018

1.9

3.61

.5252

. 752

.3950

. 960

. 5041

2.0

4. OO

.5819

. 694

. 4038

. 880

. 5120

2. 1

4. 41

. 6416

. 645

. 4138

. 810

.5197

2.2

4. 84

.7041

.599

. 4218

.750

.5280

2.3

5. 29

.7696

. 560

. 4310

. 7oo

.5287

2.4

5.16

.8380

.527

. 4416

.653

.54*72

2.5

6. 25

. 9093

.497

.4519

.612

.5565

As a check, we may obtain the ratio m con-
sistent with horizontal stability from another
point of view.

At the end of recoil, we have, neglecting
a small throttling drop, Paf*=Ph wnereas the
initial reaction, we have, AsK hence

Paf Ph
m = - = — which gives immediately the

a* v maximum ratio tor m or the
minimum ratio for r, consistent with the require-

600

nents for horizontal stability. „

Again the maximum value of m - depends

n •

further upon the heating

limitation (i. e. the permissible rise of tem-
perature of the air in the recuperator forging).
Since the question of heating depends upon the
various factors, as radiation through the cylinder
walls, the frequency of firing, etc. we must as-
sume by experience for the given type of gun, the
maximum allowable temperature use during the com-
pression of the recuperator air. Thus, using
a ratio m = 2, we have

k-l

T Paf k 0.23

— ( ) = 2 Assuming a mean temperature

Tm Pai at 25° Centigrade, we find

T - 298 * 2°'*8 = 349°,(k=1.3) therefore, the
rise of temperature during a recoil stroke becomes,
T -Tn=51°C or 92°F. This rise of temperature is
not excessive. If the rapidity of fire is great
the mean temperature will rise. The quantity of
heat generated is the work done on the air
divided by the mechanical equivalent of heat.
Now if the mean temperature has risen to its
constant maximum value, then the heat generated
during the firing stroke, must be dissipated
by radiation through the cylinder walls during
the period of loading, the temperature gradient
varying, decreasing during the process of
radiation through the cylinder walls.

Thus we see we have two aspects for the
maximum ratio of m and the corresponding minimum
value of r:

(1) To maintain at best, a constant
resistance to recoil throughout the
recoil at horizontal elevation,
rather than a rise in the over-
turning force at the end of recoil
which would be entirely opposite

601

to the requirements for the proper stability
slope at horizontal recoil. This limitation
for r is of special importance when pn~Ky is
small, we have rfflin determined by the equation,

(m-l)r = 6.87 where we may obtain • d r

ai for various values of (m-l)r*
from the previous table.

(2) On the other hand, when Ph~Kv

is large as with guns of low elevat-
ion and where we have a good margin
of stability, the peak effect in
the throttling becomes larger or
the ratio

Pv>
— IL is larger, therefore, a higher

"v value of m can be used. In such
a case we become limited by the
rise in temperature of the air
during the firing.

(a) When (2) becomes the
limitation we may use a higher
value of m and therefore a
lower value of r.

(b) The expansion of the oil
varies considerably with
temperature, different
viscosities, the rapidity or
frequency and continuity of
fire, etc. Therefore the
floating piston will have
various initial displacements,
resulting in different initial air
pressures and most important

in unsatisfactory functioning
of the buffer on counter re-
coil, the buffer action start-
ing at various different points

602

on counter recoil, unless the
ratio is made sufficiently large,
since with a large bulk of oil,
temperature changes and consequent
expansion is less.

(c) Due to the wear of the pack-
ing on the floating piston it
has been customary to limit the
maximum velocity of the float-
ing piston to not over 12 ft.
per sec. though it is believed
that the packing may be designed
to withstand a surface velocity
of 20 ft. per sec.
Therefore, in conclusion, from a consideration

of (a), (b) and (c) rmin ordinarily should be

limited to: rmin=1.3 to 2.0

In difficult designs, however, the proper

minimum value of r should be determined more from

a consideration of aspects (1) and (2) in (a) rather

than (b ) and (c).

Limitations for the maximum value of r:

For very large values of r, slight changes
in the quantity of oil due to leakage, will have
a marked effect on the relative initial positions
of the floating piston and recoil piston. Further
it would be difficult to gauge slight variations
in the quantity of oil due to the relatively small
motion of tbe floating piston which moves the
gauge rod .

But most important from a point of economy
in the weight of recuperator forgings and very
often in a satisfactory layout, a too large value
of r becomes prohibitive.

fle limit the max. value of r to, rmax=3.5
Rence the design limitations for r becomes for
an ordinary layout with 3 cylinders r =>1.8 and
r-^3.5. When only two cylinders are used we have

603

tbe following considerations:

(1) If the length of the recuperator
air cylinder has the same total
length as the recoil cylinder, we
x have ki=0.8 and ka*0, hence
k

A=0.8 A hence r =

0.8 i

Ab=0.8 A b hence r

.
mk - 1 mk - 1

If m = 2 a heating limit on the ratio of com-
pression, we have 1

k

— - - = 2.42 for k = 1.3

I

m* - 1

and

2.42

r « - = 3.025
0.8

On the other hand (in consideration of a
constant recoil reaction for stability at
horizontal elevation) if we decrease m to m * 1.3,
we have 3.73

r * • - 4.67 which gives a very
0 • o

bulky recuperator with

too snail relative displacement of the floating
piston as compared with the recoil displacement.
Therefore, if only two cylinders are to be
used and of tbe same length we are peculiarly
limited by bulk and a small movement of the
floating piston as compared with the recoil piston,
on the one hand, while with a decrease in the
ration r, the final air pressure is increased and
overbalances the peak of the throttling plus the
initial air, thereby giving a rise in the recoil
reaction towards the end of recoil at horizontal
elevation.

Therefore as a compromise, if two cylinders
must be used of the same length we may take
r = 3.5 and m = 1.8 and r is to be considered

604

constant for this combination. Thus we see by
the use of two cylinders only and of the same
length, the ratio cylinders become excessive
if aoderate compression ratios are maintained,
whereas for moderate cylinder ratios we must
maintain high compression ratios which cause
undue heating and a rise of the recoil reaction
at horizontal elevation.

A more satisfactory combination for two
cylinders only, is by use of longer recuperator
cylinder than recoil cylinder. This is usually
feasible especially for guns, where the tube is
long. A satisfactory approach to three cylinders
may be had by the use of a sufficient overhang
of the recuperator cylinder.

The air column in the overhang can be
reasonably assumed at 0.5 the horizontal recoil.
Therefore in the equation VQ=Aa(kt+k2.)b . We may
assume as before kt=0.8

k »0.5

2

= 1.3
hence kt +ks

Hence with a ratio of pressure expansion in the
recuperator m = 1.5, we have 3.73 Ab=1.3Aab hence

Aa

— = r = 2.87. By increasing the ratio of ex-
pansion we may limit the minimum
r to: *n)in=2'5. From the above it is evident
that though it is not feasible to use a recuperator air
cylinder as long as the combined length of a
separate recuperator and air cylinder, on the other
hand the minimum ratio of r is greater than with
an ordinary layout and with only two cylinders,
the total weight of t"he forging may exceed that
of these cylinders. Hence if two cylinders
are to be tried in place of three, preliminary
separate layouts for the two combinations should
be worked out in view of minimum weight and

605

satisfactory layout before either plan is adopted.
Therefore, in a design layout we start with

which determines the

Ph~kv
'a

ratio — providing it falls within the limits
A

rmin and rmax* Hence the recoil area,
becomes,

A = — —
4600

Anti-aircraft Guns;

Anti-aircraft guns are the most difficult
to design without having excessive bulky re-
cuperator forgings, Pn-Kv becomes small, since
Ky is larger to hold the gun in battery at max-
imum elevation and p^ is small to satisfy stability

at zero elevation, further ps is large, there-
fore r is in general large, usually 3 or above.

If r exceeds 3.5 using 3 cylinders, we must
increase Pt]~Kv either by reducing Ky and then al-
lowing the gun to return slower into battery at
maximum elevation and with a smaller margin of
excess battery reaction of the recuperator at
maximum elevation or preferably increase pn at
the slight sacrifice of stability at zero
elevation, - in this case we have

Ph=Kv+. 000912V8 — —
3.52

It will be rarely found, however that r ex-
ceeds 3.5.
Howitzers:

With Howitzers, we again meet the condition
of a large Kv but since horizontal stability is
not a consideration, pn may be relatively large,
and therefore pn~Kv still may be maintained
large.

606

r is generally medium or small at the sacri-
fice of horizontal stability.

Guns :

With guns, r is in general small, usually
from 2 to 3. If r decreases below, 2.0 to 1.8
using 3 cylinders, or 2.5 using 2 cylinders with
a longer recuperator cylinder than tfce recoil,
the effective area of the recoil piston is now
determined by the formula:
x

A = 0.2425 — (pu - Kw) and the maximum pres-

v*

_ sure in the recoil
P g

cylinder, becomes, pmax= —

A

Howitzers and Guns on Same Carriage:

When Howitzers and guns are adopted for t"he
same carriage, Ky must be large for the howitzer
and ph small for the horizontal stability of the
gun. Therefore pn~~Kv is small, ps large and we

meet exactly similar conditions as with anti-
aircraft guns.

Hence, for this combination, r is in general
large, usually 3 or above. If r exceeds 3.5,pj,
must be increased or Kv decreased, these values
being connected by the relation

KV+.000912V«

(7) Calculation for air pressures and volumes:

From (5) we have for the initial recuperator
reaction

Kv -

k

Wrsin

^m a

X+Ci

where k

Pmax may
at 4500

in. a
+.09(a+ s±)

B

= 1.2 to 1.3
be assumed
Ibs. per sq.

JPmax

1

C

2nkK

vdb

cz

A
+d)[.05b

^O.lSn

nr
WP

C =0.73n(d_+d)[.05b + .09(a + —)]

2

PS

A = = effective area of recoil piston

4500

a, at and b are contact lengths of the pack-
ing in inches.

d^- distance down from center of gravity of
recoiling parts to line of action of Kv
in inches.

1 = distance between clip reactions in inches
n = 0.15

r = distance down from center of gravity of
recoiling parts to mean friction line
in inches.

For guns of low elevation and reasonable stability
where r falls below, r we have

A = 0.2425 — (p - Ky) and

V

Id = /0.7854A+dr

2

r
A = 0.2425 — (ph-Kv) and the maximum pressure

^* in the recoil cylinder

is thereby reduced to:

PS
Pmax = —* and the area of

A

the recuperator

cylinder becomes, Aa=rminA. With an ordinary
layout using 3 cylinders, rmin=2 and with 2
cylinders and an overhang, rmjn=2.5. Knowing Aa,
we have for the inside diameter of the re-
cuperator,

Da :

0.7854

To determine the diameter of recoil cylinder,
we must know the area of the piston rod,Ar. Then

/A+Ar

D * /• where A_ is determined as follows:

0.7854 r

608

If 6 is the total hydraulic braking including
the joint frictions at the stuffing box and re-
coil piston, we have B+Rg=K+Wrsin# where

2ndb

)=K+Wrsin

l+2nr

The maximum stress in the rod is at a section
at the lug, at maximum acceleration of the recoil-
ing parts and at maximum elevation. If

T = the tension in the rod at the lug
L

K+Wrsin0ffiax

2ndb
1+

(1- =-)+ Pb r-

l+2nr

where We » weight of rod and recoil piston

pb = total maximum powder pressure on base

of breech.

If fDax is the maximum allowable working fibre
stress in the rod, we have f

* max

and Aa * rA. Now with guns of low elevation,
Ky is snail and ph relatively large, hence the
difference pn~Kv is large and ps is small. There-
fore r becomes small.

In (7) tables and a chart has been con-
structed giving values of m and r for different
air column lengths, the air columns being expressed

609

in terns of a ratio of the length of air column
divided by the length of recoil, that is

j « - where 1 = length of ai* column,

b

b = length of recoil.

A maximum limitation of m, based on a moderate
temperature rise of the air during the recoil and
a constant reaction throughout the recoil at
horizontal elevation (i. e. no increase of the
recoil reaction at the end of recoil, ) will be
taken at 1.8. Evidently for different air column
lengths, we will have different minimum values
of r, corresponding to a waximum value of m =
1.8.

The longer the air column lengths, the lower
the ratio r.

If r falls below the minimum allowable value
of r (i.e. the r corresponding to m = 1.8) for a
given air column length, r becomes a constant,
and the area of the recoil piston must be in-
creased according to the formula*- then

Kv
pa^ = -j- Ibs. per sq.in. initial recuperator pres-

sure intensity.
Hence the initial air pressure becomes,

Pai s Pai + ' a*d Paf

Aa

t
C||l = 1.46nda[.05b + .09(a + y)]

Paf

Next to determine the proper ratio of m = -

Pai
we must consider the following:

The initial volume Vo is expressed by either
memoer of the equation,

610

» Agl where k » 1.3,

b = length of recoil
1 » length of air column,
reduced to an
equivalent cross
section A.

Since r * — , we have
A

r - • rj, where j
b

The following tables give the relation of
o and r for various values of j.

Pax
Pai

0.8 r

1.3

.77

m

r

1. 2

9.575

1.4

5.476

1.6

4. 116

1.8

3. 442

2.0

3.002

2. 2

2.746

2. 4

2. 546

2.6

2. 401

2. 8

2. 282

3.0

2. 188

3.2

2. Ill

3.4

2.047

3.5

2.018

611

Values of m and r for j = 1.1

1. lr

1. lr

1. lr

r

1. lr

1. lr

•

log...

Klog

m

-1

1. lr-1

1. lr-1

1. lr-1

1. 8

1. 98

.98

2.020

. 30535

.39696 .

2. 494

2.

2. 2

1. 2

1.833

. 26316

.34211

2. 198

2. 2

2.42

1. 42

1. 704

.23147

. 30091

1.999

2.4

2.64

1. 64

1. 61

.20603

. 26888

1.857

2.6

2.86

1.86

1.54

. 18686

. 24291

1.750

2. 8

3. 08

2. 08

1. 48

. 17056

. 22173

1.666

3.2

3-52

2. §2

1. 4

. 14520

.18876

1. 544

3.4

3- "74

2. 74

1.36

.13513

. 17567

1.499

3.5

3.85

2. 85

1.35

. 13066

. 16986

1.4^9

Values of m and r for j

» 1.3

1.3'

1. 3r

1.3r

r

1.3'

1. 3r-i

Klog

•

1.3'-1

1.3*-1

1. 3r-l

1. 8

2.34

1.34

1. 746

. 24204

.31465

2.064

2.0

2. 60

1. 60

1. 625

. 21085

.27411

1. 880

2. 2

2.86

1.86

1. 538

. 18696

.24305

1. 850

2. 4

3. 12

2. 12

1. 472

.16791

. 21828

1.653

2.6

3.38

2. 38

1. 420

. 15229

. 19798

1.577

2. 8

3.64

2.64

1.379

. 13956

. 18143

1.519

3.0

3.90

2. 90

1.345

. 12972

.16734

1.470

3-2

4.16

3.16

1. 316

. 11926

. 15504

1.429

3. 4

4. 42

3.42

1. 292

. 11126

. 14464

1.395

3.5

4.55

3-55

1. 282

. 10789

. 14026

1. 381

-sol *i fcfl

612

Values of n and r for j = 1.5

l.Sr

l.Sr

l.Sr

r

l.Sr

l.Sr 1

Klog — — -

m

1. 8

2. 70

i.7o

1. 588

. 20085

. 26111

1. 824

2. 0

3. oo

2. 00

1. 50O

.17609

. 22892

1. 694

2. 2

3. 30

2. 30

1.435

. 15685

. 20391

1. 599

2. 4

3. 60

2. 60

1.385

. 14145

. 18389

1.527

2.6

3- 90

1.345

1.345

. 12872

.16734

1. 470

2. 8

4. 2O

3. 20

1.313

. 11826

. 15374

1. 425

3.0

4. 50

3.50

1. 286

. 10 9 2 4

. 14201

1.387

3.2

4. 8O

3. 80

1.263

. 10140

. 13182

1.355

3.4

5. 10

4. 10

1. 244

. 09482

. 12327

1. 328

3. 5

5. 25

4. 25

1.235

.09167

. H917

1. 316

Values

of m and r for j

= 1.7

1 *7»«

1.7r

1.7r

1.7r

r

i . /r

1.7r-l

log —

m

1. 8

3.06

2. 06

1. 485

. 17184

. 22339

1.672

2.

3.4

2. 4

1.4166

. 15124

. 196612

1. 572

2. 2

3-74

2. 74

1.3649

. 13510

.17563

1. 498

2. 4

4.O8

3.08

1. 3246

. 122084

. I587o

1.441

2. 6

4. 42

3-42

1. 2923

. 11139

. 144807

1.395

2.8

4.76

3-76

1.2659

. 102O5

. 13266

1.357

3.2

5. 44

4. 44

1. 22522

. O8820

. 11466

1. 302

3.4

5. 7s

4. 78

1. 2092

. 08249

. 107237

1. 280

3. 5

5.95

4.95

1. 2O20

.07997

. 103961

1. 2704

The values in these tables have been plotted,
with the chart, fig. (12).

The ordinates in this chart give values of
n corresponding to values of r in the abscissa.
Each curve represents the relation of m and r
for a satisfactory layout, with a given air
column, the air column length being expressed as
a ratio with respect to the length of recoil.

The air column lengths are taken at 0.8b,
l.lb, 1.3b, 1.5b and 1.7b where b = length of recoil

Values of m and r for air column lengths inter-
mediate between these values, may be easily ob-

613

-xo'r'* : U**v.U ,? at

614

tained by interpolation.

Solving for the proper value of r, and as-

suming an air column length, NO immediately ob-
tain ID and therefore paf since pai is now known.

To prevent a rise of the recoil reaction
during the recoil at horizontal displacement as
well as to minimize the temperature rise, during
the recoil, we will limit m to 1.8.

Therefore r is definitely limited for various
air column lengths. Its upper limit is more or
less arbitrary, it being desirable to prevent a
too bulky forging and obtain minimum weight. The
upper limit of r will be assumed at r * 3.5.

Thus, when we have but two cylinders of the
same length where the air column is somewhat
shorter than the length of recoil * 0.8b, we find r
very definitely limited to a constant value 3.5.

(8) Strength of Cylinders:

Strength of cylinders should be based on
maximum pressures. As shown in Chapter IV.

3

Dt = • Do where pt* - x elastic limit of the
Pt~P material used (Ibs.

per sq.in)

p- maximum pressure in
cylinder (Ibs/sq in.)

* Pmax usually 4500 Ibs.per
sq.in. in recoil cylinder

* paf final recuperator

pressure in recuperator
cylinder
D =outside minimum diameter in

inches.

D0- inside diameter which is given in inches,
Thickness between cylinders should be

Pd+Pafda
w * — — — — where w = minimum allowable tnick-

615

ness between cylinders (inches)

d = diara. of recoil cylinder in inches
da» diam. of air cylinder in inches

(9 ) - Calculation of maximum and minimum throttling

areas.

Since all port areas are constant multiples of
the maximum throttling area, the exact deviation
of this area is of prime importance.

From (6) we find, for the throttling through
the regulator orifice,

"h " TTT7 - Z~~\ (max. throttling area)
I/O vpn~Kv )

V* A 3TT*

w8 „ K A v (min. throttling area)
175(pa-Kv)

where wh - the max. throttling area usually at

horizontal recoil (sq.in)
ws » the rain, throttling area at max. elevation

(sq.in)
A = the effective area of the recoil piston

(sq.in)
V - the max. restrained velocity = 0.9 V_

approx.

K = - the throttling constant
0.773

pj, » the minimum pull, usually at horizontal

recoil (in Ibs)
ps =» the maximum pull, at maximum elevation.

(10) Layout of Recuperator Forging:

In the layout of a recuperator forging, we
must decide as to the arrangements of cylinders.
Depending upon the value of "r"* _^a_ , we have
three possible arrangements:

4 o jr. pi'SKfT'O r ' s< j

616

(1) Three cylinders, the two re-
cuperator cylinders symmetrical with
the recoil cylinder.

(2) Two cylinders, the recuperator
cylinder having an overhang with
respect to the recoil cylinder.

(3) Two cylinders, the recoil and re-
cuperator cylinder being of the same
overall length.

Paf
From the chart giving values of n = for

values of r = *a for different air Pai
colunn

lengths, we see that the values of "r" for the
curves giving lengths of various air columns are
limited on the one hand by the maximum value of
m = 1.8 consistent with stability at zero degrees
elevation and normal use of temperature in the re-
cuperator, thus giving the various mini Bum limit-
ing values of "r" for vari ous air columns, while
on the other hand the maximum allowable value of
r = 3.5 depends upon proper counter recoil function-
ing and layout considerations. If now r is obtained
by the formula,

r * .0309

The possible lengths of air columns consistent with
the limitations are determined.

If r from the above equation is low, then
we must have longer air column lengths and there-
fore usually three cylinders, whereas if r is large,
short air column lengths are possible and two
cylinders may be used. With arrangement (3), r

becomes practically constant and unless r from the
above equation falls in the neighborhood of 3,5,
it will be necessary to increase the effective
area of the recoil cylinder with a consequent
larger recoil cylinder.

617

Having decided upon the arrangement and
number of cylinders from a consideration of the

proper air column consistent with "r" we have now
to obtain the exterior dimensions of the forging.

Exterior Dimensions:

The primary exterior dimensions of importance
are:

(1) A cross section of the recuperator,
giving the location of the piston rod
with respect to the center line of

the bore, the axis of the several
cylinders, and the position of the
guides, thus determining the external

8-iv . } - •- if • ^*v -> - - ' T»;

contour of the cross section of the
»d V,*n lu

forging.

(2) A longitudinal section of the
recuperator, giving the overall
length, location of the trunnions,
elevating arc, etc.

In a satisfactory exterior layout, the follow-
ing points must be observed:

(a) The center of gravity of
the recoil parts should be

made in a vertical plane through
the axis of the bore and at a
minimum perpendicular distance

below the axis of the bore con-
sistent with a satisfactory
layout.

(b) The center line of the en-
trance channel to the regulator
valve, (that is for the passage-
way between the recoil and re-
cuperator cylinders) should
pass through the center of the
recuperator cylinder. Preferably
the center line of the entrance
channel should be in a horizontal

618

(c)

plane.
If <

the connecting channel cross
section, and D the diameter
of the recoil cylinder, the

distance between the center
line of the recuperator
cylinder and recoil cylinder
must not exceed D
To meet condition
(a) the recoil axis is
usually nearer to the axis
of the bore.

To overall lengths of the
recuperator forging nay be
estimated roughly from the
following table:

aa inches.
2

4. 7"-M« 19O6 94" "TO"

4. 7 "-«. 1918 68.75" 40

3.3'-w.l913 86* 60'

75 »/»-M.19l6 72.83" 46'

1.52

Therefore ordinarily the total length of re-
cuperator forging trill be taken at 1.5 the length
of max. recoil. It should be shorter if
practicable.

(e) Without a balancing gear,
for guns of moderate elevation,
the trunnions should be located
•in the horizontal direction at
the center of gravity of the
tipping parts plus one-half
weight of projectile and charge

619

when the gun is in battery.
More or less error in the
location of the trunnions as
respects the center of
gravity of the tipping parts
in the vertical plane per-
pendicular to the axis of the
bore will not effect the
balance, unless the angle of
elevation is considerable, and
the center of gravity of the
tipping parts is considerably
above or below the trunnions.

Therefore, in order to prevent

% : T- j , r

a reversal of the reaction on
the elevating arc and pinion
during recoil and counter re-
coil, it is highly desirable
with guns of moderate elevation
to locate the trunnions on or
below the center of gravity
of the recoiling parts which
are usually below the axis of
the bore.

When a balancing gear is introduced, as is
sometimes necessary when the gun fires at high
elevation, the trunnions are placed axially or in
a longitudinal direction, farther to the rear in
order to have as long a recoil as possible at max.
elevation. Further with a proper design of the
balancing gear location of the trunnions with
respect to the center of gravity of the tipping
parts in a direction perpendicular to the axis
of the bore is no longer so restricted except that
in order to avoid reversal of stresses on the
elevating arc it is desirable to locate the trunnions
slightly below the center of gravity of the re-
coiling parts, but the distance must be quite small
or the arc reaction will become large.

620

In gone designs it may be necessary to locate
the center of gravity of the recoiling parts above
the bore, and the ponder pressure couple will then
be in the opposite direction.

If P^e is the powder pressure couple, and K
the resistance to recoil and S the distance down
from the axis of the bore to the center of the
trunnions, in order that there be no reversal of
stress on the elevating arc, we oust have

(Pb-K)e

K(S+e) = > Pbe hence S * >

In K

determining the final values of e and S, the weight
components, out of battery and conditions existing
in counter recoil must be considered.

(f) With guns above 155 m/m,
two separate recoil systems

*% L d "'•

symmetrically placed above
and below the gun should be
used. The gun should recoil
in a sleeve and the trunnions
should be located slightly
below the axis of the bore.
Interior Dimensions;

The primary interior dimensions of importance,
are:

(1) The port area or channel leading
from the regulator towards the
floating piston in the recuperator
cylinder should bear a constant
ratio to the maximum opening of the
valve which occurs for minimum pull,
usually at horizontal elevation.

If «c » the constant channel area from the

regulator valve
wh » the max. recoil orifice.
Then wc = 4.3 wh

(2) The area of the channel or port
connecting the recoil and recuperator

621

cylinder, wa should have the fol-
lowing relation with respect to
wjj, that is, wa =* 3.5 to 4.3 wb

(3) The entrance channel to the
regulator valve froa the recoil
cylinder a, which is also the area
at the base of the regulator
valve, should be:

a = k «h where the limits of h are 2.3 to 3.5

If we pass a cross section of the recuperator

through the center of the regulator valve, the

channel area on either side of the valve, that

is

wc -(the vertical section through the axis of the

valve normal to the recuperator axis enclosed

within the area wc )= w^

and w
» -> c

If h represents the depth of the section wc
and da the diameter of the regulator valve at
its base, we have roughly,
wc - dahc = 0.5 wc = 0.55 wcapprox . Hence

"c wb "h

da ' 0.46 = 1.935 — and a = 2.94

bc hc h*

Mow in a suitable layout hc - 0.2 Da where Da »
the diameter of the recuperator cylinder, hence

-&

a - 73.5 —
a

(4) The length of the buffer chamber
s c: «*;••;••-. -:i - • '« tflk

_n. ^ ,. is based on a consideration of

-.•:•

counter recoil.
If dfa = the recoil length during the buffer action

d£ = the length of the buffer
From a consideration of counter recoil,

622

7 »rV*
db(0.15 wr+Rp)» If V - 3.5 ft. per sec.

0.8 as a maximum value, we
0.238Wr

have db* — — — — -
0.15Wr+Rp

*nd 0.238Wr A

db' * ( ••• ) -T— * min* length of buffer

where Rp-0.15 K (d+dr)[ .05* + .09(a

and

pmax=4500 usually. The buffer chamber should
be made d£=1.2 to 1.3 d^

(11) Regulator Dimensions:

Referring to fig. (11) let,

a = area at base of regulator valve (sq.in)

at= area of upper and lower valve stem (sq.

in) •*«<

da» diameter of regulator valve at base (in)

^•ta diameter of regulator valve at stem (in)
c - effective circumference at base of valve

(in) wfi
From (10) we find that, a - 73.5

and „ D*a

h
da * 9.675 -- where wh » maximum throttling

opening (sq.in)

Da » diameter of recuperator (sq.in)
Now da *0.6da approx. and at»0.7854d| =0.2825d|

hence at»0.36a.

The opening of the valve is the effective lift

multiplied by the effective circumference of the
valve at the valve seat. Extension guides or "flaps"
to ensure proper seating of these valves reduce
the effective circumference at the valve seat.
It is customary to use three flaps of a circumferent-
ial length each, equal to the arc of 60° angle, de-
creased by two millimeters on either side, making
the linear length of flap atthe circumference equal
to the arc of 60° minus 4 millimeters. Hence

623

nda 12

c * — - + "

2 25.4

-0.3925da+0.4725

In the throttling or lower valve and its stem
equalizing pressure ports should be bored within.
In the stem itself, the inside diameter or
diameter of the vertical port should be

dl »0.5 to 0.6 d.
t *t

Pour equalizing holes just above the seat in
the regulator valve, in a horizontal plane, meet-
ing at a common opening at the center should be
inserted. From the center opening there should
be a very small vertical opening leading to the
recoil cylinder, this acting as a pressure
equalizer between the recoil and recuperator
cylinders. The opening, however, should be made
negligible as compared with the throttling open-
ing and small as compared with the counter recoil
constant orifice.
(12) Reactions on Regulator Valve:

Let Pb= reaction of Belleville washer on

regulator valve (in Ibs)
< reaction of spiral spring on regulator

(in Ibs)
p = pressure in recoil cylinder (Ibs/sq.

in)
pa= pressure in recuperator (Ibs/sq.in)

a = area at base of valve (sq.in)

a - area of valve stem (sq.in.)

h * lift of valve from initial opening

(in)
h0=lift of valve from seat of initial

opening (in)
c = effective circumference at base of

valve in inches.
Sb= spring constant of Belleville washer

(Ibs/in)
Sg* spring constant of spiral springs (Ibs/

in)

624

hjj = initial compression of Belleville washer

at initial opening (in)
h ^initial compression of spiral spring at

initial opening (in)

Then, at short recoil, or intermediate recoil, we
have pa -p^(a-at )=Fb+Rs (approx ) hence

(p-pa)a+PaatsRb+Rs <!>

and at long recoil, we have

(p-pa)a=Rs(approx) <2)

• : .• /•

Further, we have the following lifts of the
,
valve,

K A V — (3) At short recoil

I loom

U) At long recoil

where ps and ph are the values of p at short and
long recoil respectively,

0.773

The spiral spring should be designed on the
following basis:

(1) The maximum compression should
be taken at from 2/3 to 3/4 the
solid load of the spring.

(2) The initial compression should
be taken at from 1/4 to 1/3 the
solid load on the spring. Hence,
using the maximum limits, the
compression from free to solid
height hfs=2b (5) and
therefore ,f n«

S3

h = - n (6)

where f83 max. allowable torsional fibre stress
(Ibs/sq.in) (Usually = 120,000 Ibs.per
sq.in)

625

Ds= diam. of helix in inches.

dg= diam. of wire

N = torsional modulus (taken at 12,000,000
Ibs/sq.in)

n = number of coils of the spring

Proa previous design layouts, the total height
of spring column, at assembled height should not
exceed pa inches. Hence the solid height H0

2°.

becomes, — - - h = HQ but H0=dg(n+l) hence

0.5Da-h

d = - r^— (7)

n+1

Combining (6) and (7) we have
nfD 0.5D-h

n

(8)

2N" h n+1

The load at assembled height = - Rg

9

hence 3nf,d|

R = - ^— (9)

32D
Combining with (7), we have

3nf_ 0.5Da-h

and with,

nfa D«n 0.5D -h

(Eq.8) we may determine

2Nh n + 1 , _.

n and D. The

solution may t>e simplified by assuming

-, as a first approximation, we
u art* lo aoUo»

0.5Da-h 0.4Da

^^— — —— •= — — —

0+1 n then have

fsDt
RS« .01885

0.8 Da

626

Solving for D, d,and n, we have

2.62 / — : inches

If we assume N=10,500, 000 Ibs/sq.in.

fs = 120,000 Ibs/sq.in.
then

0.216

If D is too great for a satisfactory layout,
we may increase the height of the spring column
slightly or let the maximum working load on the
spring move closely approach the load at maximum
compression.

Solving for the diameter of wire "d" and the
number of coils "n", we have
s

and if fs=120,000 Ibs per sq.

in. then
d = .0305 y R,,DS

o o

Test pressures are usually at double the
service pressure, hence the material will be
strained up to 3/4 the elastic limit.

(13) Design of Cam Mechanism and Layout;

Briefly, the action of the cam is to control
the motion of the upper valve stem which reacts
against the Belleville washers. At long recoil
the valve displacement (i . e. the displacement of
tbe unoer valve stem) is sufficient so that the
lower valve stem, is never brought into contact
with the upper stem, and the lower stem is controlled
entirely by the spiral spring. At an intermediate

627

recoil, the lower stem is brought ultimately in
contact with the upper stem and the valve is con-
trolled by the compound characteristics of the two
stems. As the upper stem initial position is
brought closer and closer to the lower valve stem,
the valve opening depends more on the characteristics
load deflection slope of the Belleville washers.
Finally at short recoil, where the upper valve stem
is brought into initial contact with the lower valve
stem and the displacement of the cam is zero, the
valve opening depends practically on the Belleville
characteristics alone, the effect of the spiral
spring being negligible. It is to be noted that
the throttling at intermediate recoils approximates
that if a constant orifice, with however the
characteristic peak effect in the braking with a
constant orifice eliminated. The throttling, there-
fore depends upon the displacement of the valve
and the characteristic load deflection curves of
the Belleville and spiral springs.
Let g = ratio of cam movement to valve movement
(usually taken at 5)

X = distance valve should lift to engage

Bellevilles (in)

hs- initial compression of spiral springs (in)
bo = clearance of valve (in)
h = lift of valve (in)

hb3 initial compression of the Bellevilles (in)
Sb3 change in load per unit deflection of the
Belleville washers, i.e. the Belleville
spring characteristic (Ibs)

Sg* change in load per unit deflection of the
spiral spring, i.e. the spiral spring
characteristic (Ibs)

Then at an intermediate recoil, the reaction of
the spiral spring, becomes, Rg=Sshs+Ss (h+ho)(lbs)
The reaction of the Belleville washers becomes
Rb=Sbbb+Sb(h+h0-X) (Ibs) while the hydraulic re-
action becomes, (p~Pai)a+Pai at (Ibs)

628

where p = the intensity of pressure in the recoil
brake cylinder (Ibs/sq.in)

pai = the intensity of pressure in the re-
cuperator (Ibs/sq.in)

a = area at base of valve (sq.in)

at- area of valve stem (sq.in)
Then for equilibrium of the valve,
Ss(hs+h+h0)+Sb(hb+h+h0-X)-[ (p-pai>a+Paiat}* °
Therefore, for the distance of valve lift to engage
Bellevilles, is

]>
J

X = — JsgOvho+hJ+Sba^+bo+hM (p-pai)a+paia

Sb L

The variation of the length of recoil against ele-
vation may be made in any arbitrary way, but, how-
ever, the following method is usually employed.

In general, assume the le ngth of recoil that
of horizontal recoil from 0Q to #t degrees, (usually
from 0° to 20° elevation), then, decrease the re-
coil proportionally with the elevation, (i.e. from
20° to max. elevation, the recoil length decreases
uniformly to short recoil at maximum elevation).
Thus if,

b = length of an intermediate recoil
0 * corresponding angle of elevation
bh= length of recoil at horizontal elevation
bs= length of recoil at maximum elevation
Of Q- maximum elevation
0^= initial elevation where the recoil is

shortened
. ,
then b b

b = - «?.-0)+bs (ft)
0m-<*i

The resistance to recoil corresponding to the
length of recoil "b" is given by:

K = 1.03[

2g uVf
b+ (.096+. 0003d)

a u

629

where * - weight of projectile (Ibs)

W » weight of powder charge (Ibs)
* Wr= weight of recoiling parts (Ibs)
u = travel up bore (inches)
d = diam. of bore (in)
v * muzzle velocity (ft/sec)
Vf* max. free velocity of recoil (ft/sec)

and *v+W4700

Vf» - (ft/sec)

*r 0.47WrV|

For a rough approximation, K = — — — — (Ibs)

gb
The required recoil braking is given by

(K+Wrsin

B = - — Rn or B = — - - — -- R_
2ueb l+2ueb

approximately
where 1 = distance between guide clips (in)

eb 3 distance from center line of bore to

center line of brake cylinder (in)
r = aean distance to guide contact (in)
R = brake cylinder packing friction (lb»)

For the lift of the valve, we have

a

.098A*V,.

V.in) where A = the ef-

fective area of
recoil piston (sq.in)
Kv = recuperator reaction (Ibs)
Vr = velocity of retarded recoil, about 0.9Vf

(ft/sec)

c = effective circumference of lower stem (in)
we may also express the lift in terms of the
pressures, then >098A VF

h * - Un)

where p = - a the pressure intensity in the brake
A cylinder (Ibs/sq.in)

630

SPRING* & CAM MOTION (x) DIAGRAM
Fig. 13

631

Ky

Pai= -— = the initial recuperator pressure
intensity (Ibs/sq.in)

(14) Counter Recoil Design:

The function of the counter recoil buffer is
to reduce the pressure in the recoil cylinder to
a very low value practically zero. The recoil-
ing parts are therefore Drought to re^st by the
combined packing and guide friction in a displace-
ment corresponding to the buffer length in the
recuperator cylinder. For a preliminary design
layout, the entrance velocity into the buffer may
be taken at a counter recoil velocity of 1 meter
or 3.28 ft/sec, but preferably less than this. To
allow for a margin in variation of counter re-
coil friction the buffer displacement will be re-
duced in the counter recoil to 0.7 its actual
value. Then, we have

0.7db(0.l5Wr+Rp-)= £ ^-^ 3728* hence
0.238Wr

The corresponding displacement of the buffer in the
recuperator, is

A

— db where A = effective area of recoil piston

*a Aft» cross section area of recuperator

cylinder

The length of the buffer rod will be male about
20* greater. Hence for the length of the buffer
rod, we have

0.238W- A

The length of buffer chamber is usually constructed
from 20 to 30* greater than the buffer rod, hence
d£ - 1.2 to 1.3 db' (ft) for length of buffer chamber

632

The ID ax i BUB allowable counter recoil velocity at
borisontal elevation should not exceed 3.5 ft/sec.
The counter recoil velocity for a satisfactory

design ranges from 2.5 to 3.5 ft/sec. The velocity
used in counter recoil should be such that with
the expression g

0.7 ib<0.16Ir.B{>. ^ 55=15..

db ranges from 1/4 to 1/3 the short recoil bg

The packing friction for the recoil may be
expressed by the relation, R -C^+C^p (Ibs) where
p = Ibs/sq.in. in the recoil cylinder. On counter
recoil during the buffer action p = 0 approx. hence
R'»Ct approx. Now Ct is that part of the packing
friction due to the Belleville compression of the
packing and is designed for the maximum recoil
pressure paax (Ibs/sq.in)

If Dr= outside diameter of packing ring, (in)

dr= diameter of rod (in)

a * depth of silver flange of packing (in)

a1 « depth of outer silver flange (in)

b= packing contact (in)
then t

R«Ct* R(Dr+dr)[.05b + .09(a+ — )]0.15 p>ax (Ibs)

The guide friction on counter recoil may be taken
at RJ-0.15 to 0.2 Wr (Ibs)

For the total recoil friction, we have

Constant Orifice Opening: at max, elevation,

* A« V

(sq.in)

13.2/paA-Rf-Wrsin<Zf.

- at horizontal elevation.

633

Pai+Paf
where p^ = - - - = mean air pressure (Ibs/sq.in)

2

Pal = initial air pressure (Ibs/sq.in)
pafj final air pressure (Ibs/sq.in)
The orifice to be used should be taken the mean

of WQ and w" hence ••+•"

o o ( . ,
w0= -£— (sq.in)

and(

2»5 to 3

KAV

V0f=2002.5 (ft/sec)
The buffer entrance area should be

/ ' *£

.00894 • •
Pi - H A V

•here V = 2.5 to 3.5 ft/sec.

DESIGN PROCEDURE FOB ST. CHAMOBD RKCOIL

Given:

Clam. of bore(incbes) d =
Muzzle velocity (ft/sec ) v
Wt.of chargeCin Ibs) if =

Travel of shot up bore(inches ) u =
Max. angle of elevation 0m =
Min. angle of elevation 0 =

Max. ponder pressure on base

of "breech(lbs/sq.in) Pb = 24000 Ibs/sq.in,

Length of recoil at max.

elevation (ft) be - 2.5 feet

634

Lengtb of recoil at 0°
elevation (ft)

3.75 ft

w » weight of charge (Ibs)

M = weight of projectile (Ibs)

Wr * weight of recoiling parts

W

(los)

(Probable weight of total
[mount
[tfeight of trail

3.25 Ibs,
33. Ibs,

1260.

2700.
300.

Ibs

Ibs

Ibs

3000.

STABILITY LIMITATIONS.

Max. free recoil velocity
(ft/sec)

wv+w 4700

Max. recoil reaction
R(approx) =

wr v«

0.45 — — =

« bs

33.xl500-t-470Qx3.25
1260

51.50 ft/sec.

0.45

1260 x 51.50

32.2x2.3
18,700 Ibs.

Height of axis of bores
above ground (assumed)
(ft) h =

3. ft

Max . allowable horizontal
recoil (ft)

bn max

Max. velocity of con-
strained recoil (ft/sec)
Vr*0.9 Vf(approx) »

2x32.2

11.1 ft.

0.9 x 51.5 = 46.4 ft.

635

Recoil constrained energy
(ft/lbs)A =

1260 x 46.4
2 x 32.2

42,000 ft/lbs

Recoil displacement
during powder period
(ft) "

, ,H+0.5*, ,,33+0.5x3.25,80

Er= 2.24(— -) u = 2'24<-I^6- ->Ia -

0.41 ft.

Constant of stability
(assumed)

Overturning moment
Stability moment

Horizontal distance from

spade point to line of 2700x81+300x34

action of W3(ft) lg = 30QO ' = 6.35 ft.

0S = angle of stability 20°

d = moment arm of over-
turning force =
ht cos ef+ds-1 sin <t = 36x0.9397+7.5-81x0.3420 =

13.60 in.

Horizontal recoil con-
sistent with stability
(ft)

Wgls+WrE cos 0 +

bn a -
2Wp cos fl

/(W8ls+Wr

E cos

dA
4Wrcos(2f(WslsE )

3000x6.35x41-

42000x1.13
.96

2x1260x0.9397

636

bb >ax 3.74 ft used
bhsax.3.75 *

3.74 ft

APPROXIMATE DimiBIOi OF RICOPIRATOH
FOROIB08:

Max. resistance to re-
coiKat max. elevation)
(Ibs)

K.

Min. resistance to re-
coil (at horizontal

elev.Mlbs)

Max.pull (nax.elev. )
Ubs)
P,-K8+»rsin 0-2R

2R*Wrsin 0m(approx)

0.46 1260
2.6 32.2

18700 Iba.

51.5

0.47 1260
3.75 32.2

13100 Ibs.

18700 Ibs.

51.5

Min.pulKO8 elev.)
(Ibs)
Pb-Kh-0.3Wr *

Initial recuperator
reaction (Ibs)
Ky - 1.3Wr(sin0m+0.3

cos 0B) »

1.3"1260(. 9848*. 05"0. 1736) =
1700

637

Ratio of recuperator
cylinder area

Effective area of re-
coil piston

Aa
r S-A =

18700

= 0.039x46.35

12730-1700

2.35

From chart - assume air
column = 1.36

r»in « 2-5

Total weight of recoil
piston and rods (Ibs)

30 Ibs

Effective area of re
coil piston (sq.in)

If r > r

min

A. = 0.243

ff

Corresponding max.
pressure (Ibs/sq.in)

PS
P.ax - =

Approx .max. tens! on
rods at horizontal
(Ibs)

¥

18700+1260x0.3420+
.30
24000 Xl3'4

27000 Ibs.

Assumed max. fibre
stress (Ibs/sq.in)

max

» I elastic

limit

60,000 = 40,000 Ibs/sq.in

638

Area of recoil rod (sq.in)

max

Diani. of recoil rod

dr » 1.127

Total area of recoil
cylinder (sq.in)
Ar « A+Aa *

Inside diam. of recoil
cylinder (inches)
Dr=1.127

Area of recuperator
cylinder (sq.in)
A. = rA *

27000
40000

.676

1.127 /.676 = .925 in,
use 1 inch.

4.16+0.676 = 4.836 sq
in.

1.127 /4.8S6 » 2.48
inches.

2.5 x 4.16 * 10.40 sq,
in.

Inside diam. of re-
cuperator cylinder(in)
D= 1.127

1.127 /10.40 =3.63
inches .

COMPUTATION OP PACKING FRICTIONS,

Recoil friction

Width of leather contact
of packing (assumed ) (in)
b » 0.18 in. to 0.25 in.» 0.21 inches

Depth of one silver
flange of packing cup
(in) a * 0.14 in. to 0.16
ia. -

0.14 inches.

639

Depth of outer silver

flange

a1 = 0.18 to 0.22 in.=

0.18 inches

Constant spring com-
ponent of total pack-
ing friction (Ibs)
C - w(Dr+dr)t.05b+.09

(a +

J

n (2. 48+1)0. 05x0. 21+0. 09
(0.14+0.09)0.15x4500 =
230 Ibs.

Pressure constant for
fluid pressure com-
ponent of total pack-
ing friction (Ibs)
Cf= n(Dr+dr)[.05b+.09

( a + |A)]0.73 =
fi

n (2. 48+1) [0.05x0. 21+0. 09
(0.14+0.09)]0.73 -
0.250

Total recoil packing
'friction (Ibs)

230+0.250
Ibs.

x 4500 = 1350

Floating Piston Friction

Constant spring component
of floating piston
friction a

CJ«1.12nDa[ .05b
Paf s GPaf G *

Pressure constant for
fluid pressure com-
ponent of total packing
friction (l&s) C'=1.46nD.

a «
[.05b+.09(a+ -;)] *

0

. 63f0.05xO. 21+
0.09(0. 14+0. 09)]paf -
0.4Paf

1.46x«x3.

0312 »0.52

640

CALCULATION OP THI D I H K » 8 I 0 N 8 OF TUB

RECUPERATOR FORCING:

Max. resistance to recoil
(•ax .e lev at ion) (Ib s )

uVf

bg+ (.096+. 0003d

v

1.05

1260*51.50
64.4

2. 5+ (.096+0. 0003*4. 134)

19,700 Ibs,

80*51.5

1500

Win. resistance to recoil
(horizontal elevation)
(Ibs)

- *rVf
h 2g

uVf
bb+(.096+. 0003d)

1250*51.5

64.4 3.75+(0.096

Maxinum recoil packing
friction (Ibs)

packing friction)

Kg

(Paax»4500 or —
approx. )

+0.0003*4.134)

12900 Ibs.

80*51.5
1500

230+0.250*4500*1360 Ibs.

641

Distance between clip re-
actions (inches ) (assumed)
1 -

»-;

60 inches

gravity of recoiling
parts to mean friction
line (inches) r1 =

Distance from center of
gravity of recoiling
parts to axis of piston
rod(inches) d =

6.5 inches

Coefficient of guide

friction, n=0.1 toO.2 = 0.15

7.5

Pull at max. elevation
(Ibs)

""I*

p max

l-2nr

Pull at horizontal
elevation (Ibs)

19700+1260*0.8848

2x0.15x7.5
1+

60-2x0.15x6.5
- 1360 «• 18800

2nd;

l-2nr

Ct hence

w i"

12900
2x0.15x7.5

60-2x0.15x6.5
11,550 Ibs.

- 230

Excess recuperator bat-
tery reaction constant
(recuperator constant)
K=l.l to 1.3 =»

1.2

642

Recuperator reaction in
battery at max. elevation
k(Wrsin0m+Ct)

1.2(1260x0.9848+230)

1.1. 2(

2x0.15x7.5
60+2x0.15x6.5
= 1980

4.16

Max. restrained recoil
velocity(f t/sec)

Vr= 0.92 Vf =

Patio of

Recuperator area

Effective recoil piston
area.

47.40 ft/sec

1.625 Vr /-

2. 625x47. .

/ 18800
4500(11550

2.6

- 1980

If r < rm^n(see chart and

assume air column) 51.6 inches

Effective area of recoil
piston(sq.in)

Ps 18800

A = * — — * 4.18 sq.in.

4500 4500

Area of recuperator

cylinder (sq.in)

Aa = r A = 2.6x4.18 = 10.87 sq.in,

If r < rBin(8ee chart and
assume air column)

643

Effective area of recoil
piston

0.2425

vr

If r > 3.5(Ti»o short
cylinders - see chart)

Effective area of recoil
piston(sq.in)

Ps

A = «oo ^f

Area of recuperator
cylinder (sq.in)

A, = 3.5 A

•a

Horizontal recoil pull

Ph=Kv+. 000912V' — 2 — »
12.25

Where length of Air column is assumed

Length of air column

in terns of length of max.

recoil (assumed)

1.445

Ratio of

final air pressure
initial air pressure

M »( . ,)* ' (see chart of

rj tables) = 1.5

644

Initial recuperator
pressure (Ibs/sq.in)

v 1980

— » * 473 Ibs/sq.in.

A 4.18

Final recuperator
p re ssure( Ibs/sq.in)

Paf * Pai (approx) = 1.5 * 473 » 710 Ibs/sq.in,

Initial air volume (cu.in)

*~Al = 10.87x51.6 = 580 cu.in.

When ratio of final to initial air pressure is
assumed.

Assume

Paf Paf

m = = — r- = 1.5 = 1.5

Pai Pai

Initial air volune(cu.
in)

0.77 —0.77

Vo - (m".77_ 1 >A. bh» (4=^^)10.87x45

560 cu.in.

Length of air column
(inches)

560

a

51.8 in.

A, 10.87

Initial recuperator
pressure (Ibs/sq.in)

' . ~ » i222 , 473 Ibs/sq.in,

4.18

645

Final recuperator pres-
sure

Paf

4.73 » 710 Ibs/

sq.in

INITIAL AND PIMAL AIR PRESSURE AHD
AIR VOLUME.

Initial recuperator
pressure (Ibs/sq.in)

473 Ibs/sq.in

Floating piston
friction(initial)(lbs)

Ct+C» Pai =

.4x710 +0.52 x 473
530 Ibs.

Drop of pressure across
floating pistondbs/
sq.in)

[*cs Pai

ai

Final air pressure
(Ibs/sq.in)

Paf = rap z

Final air volume (cu
in)

z V -

o - A bh =

473

530

10.87

3-,<m a I

52° lbs/

sq .in,

1.5x520 = 780 Ibs/sq

in.

560 - 4.18 45 » 370
cu.in

646

Average drop of pressure
across floating piston
Qbs-sq.in)

Strength of cylinders

ianiS

.4x780+0.52(780+520)0.5
10.87

» 620 Ibs.

Test Pressure 2 X
Service pressure.

Area of recoil cylinder
Ar=Ar+ar =

Diameter of recoil
cylinder(inches)Dr

1.127 / Af =

Diameter of recuperator
cylinder (inches)

Da - 1.127 / Aa =

4.18 + 0.79 = 4.97
sq.in.

1.127 /4.97 = 2.51
i nches .

1.127 / 10.87 = 3.71
inches .

Max. allowable fibre stress
for cylinders (Ibs/sq.in)

Pt - - elastic limit =

Dro • Dr

- 60000 = 22500

Ibs/sq.in.

Min. outside diam. of
recoil cylinder(incbes )

2.51

22500 -t-4500
22500-4500

3.07 inches.

647

Min. outside diam. of
recuperator or air cylinder

3.71
Pt~Paf 22500 810

3.84 inches

use 3.96 inches.

Min. width between ad-
jacent cylinders
(inches)

w

PmaxDr*PafDa 4500x2.3.05+810x3.71

1.5 pt 1.5 x 22500

= 0.393

Calculation of max. and min. throttling areas

Max. throttling area
(at horizontal recoil)

(sq.in) I 1

.098A*Vr 0.098 * 4.18 x47.4

*h S y 1

(11550-1980)*
= 0.374 sq.in.

Min. throttling area at
elevati

.098A*Vr

max. elevation(sq.in)

(18800-1980)'

= 0.282 sq.in.

„* «aia Isc

LAYOUT OF RECUPBBATOR FORGIMG, PORT
AND CHAKKEL AR3AS.

. of cylinders =

648

Overall length of forging

(inches)

lf-1.5 bh » 1.5x45 » 67.5 in.

Diameter of recoil
cylinder(inches) Dp » 2.51 in.

Diameter of recuperator

air cylinder(incbes)

Da - 3.71 in.

Length of air column

(inches) la » 51.6 in.

Area of connecting
channel between recoil
and recuperator
cylinder (sq.in)

w. » 3.5 to 4.3 wh = 4.02x0.374 = 1.5 sq.in.
"

Diarn. of connecting channel
(inches)

dl » 1.127 /7T » 1.127 ifi = 1.38 in.

Max. depth of recuperator
cylinder below recoil
cylinder (inches)

da 1.38

0' »D. * 2.51 » 1.82 in.

2 2

Constant channel area
from regulator valve
(sq.in)

•c » 4.3 wh = 4.3x0.374 = 1.606 in.

Depth of constant chan-
nel area Nc(inches)

hc » 0.2 Da = 0.2x3.71 = 0.742

Extreme area to regulator
valve (sq.in)

a - 73.5 -

649

Diameter of entrance channel
(inches)

da » 9.675 — =

Uo

9.675

in

PK8I8K OF B10ULATOB.

Area at base of regulator
valve (sq.in)

"n
a * 73.5

D:

Diam. of regulator valve
at base (inches)

*h
d. = 9.675 ~ =

8 Da

Diam. of upper and lower
valve stem (inches)
da = 0.6 d0 =

Cross section area of
upper and lower valve
stem(sq.in)
at = 0.36 -

Diam. of inside port in
valve stem

dl=0.5 to 0.6 d- -
a

0.747 in.

0.975 in.

0.6 x 0.975 = 0.585
in.

0.36x0.747 = 0.269
(sq.in)

0.66x0.585 » 0.322
(sq.in)

Guides or flaps at base of
valve use =3 subtends are
60°- 2mm on either end.
Length of one flap at base
of valve c1 = 0.524 da -0.
1573 »

0.524x0.975 -0.1573=
0.354 in.

Effective circumference at
base of valve (throttling
area) (inches)
c * 1.571 da+0.4725 »

1.571x0.975+0.4725=
E.004 in.

660

Load on spring and
Bellevilles at short
recoil (max. elevation)
(Ibs)

Load on spiral spring
at long recoil -(0°
elevation) (Ibs)

Lift of valve (inches)
short recoil(max.
elevation).

.098A2 V,

(4500-473)0.747*473x0.269
=* 3140 Ibs.

4.18
1710

- 473) 0.747 =

h1

Lift of valve(inches )
long recoil(0° elevation)

3

.098 A* Vr Vh

h" * y - — *

°/Ph-Kv

Load at solid height
on spiral spring libs)

R»« * r R. s

0.282
2.004

0.1405 in

0.374
2.004

0.1862 in.

1710 - 2280

651

Spiral regulator spring

iil?J!^3

vfax . tor sional fibre
stress (Ibs/sq. in)

100,000]
fs = 120,000 > =

140,000 J

Torsional modulus
(Ibs/sq.in)

:i?.f* y

11,500,000 1
N = 10,500,000 f =
10,000,000 J

SeS ^fii

-vyfc ri^i

H5i ilOO*

Diam.of helix
spiral spring
regulator valve

1 * C *

pinches; »

f& 3

/N*h"^R

; i 't) bo'

io ci-tgn;

Ds- 4,.G^/ •; 8

£<

If N = 10,500,000
fs = 120,000
(Ibs/sq . in )

A"*SS
D =0.216 / =

a 3

D!

Diam. of wire of spiral
spring (inches )
3/CD7
i =1.503 7 ? ? =

fs

If fs =120,000 Ibs/sq.in.
i =.0305 •/R3DS

- • 5 c c? .

' -% A J "fcT
"-*•*/ — 1 9w

. «

652

COUETiK BICOIL DgSIGH - BUFFER DBSIGH
COHSTAMT ORIFICg AMD PORT
AREAS.

Packing friction at end
of counter recoil (Ibs)

23° lbs

Recoil length during buffer
action (ftl

0.238Hr 0.238x1260

db * r> isw +R'° - = °-715ft

0.15Wr+Rp .015^1260+230

l!L"'to'bs) _8-61n-

Length of buffer rodlft)
0.238W

. A

= 1.2 - - f =
0.15Wr+R^ Aa

2 db 1.2x.715

= - = .33 ft.=

r 2.6

3.96 in.

Length of buffer chamber
(ft)

d£»1.2 to 1.3 db' = 1.25 x 3.96 = 4,95 in.

Win. allowable counter
recoil velocity (ft/
sec) at max. elev.

Max. allowable counter
recoil velocity (ft/
sec) at horizontal
elev .

v0. * 2.5 - 3.5 ft/sec? 2.5 ft. sec

653

Total counter recoil friction
(nax. elevation) (Ibs)

W_(sin0_+0.3cos0m)

+0.3Wrcos0m

230+.25[

1260('9848*

4.18

.3X.1736,

] +.3xl260x

.1736 = 290 Ibs.

Total counter recoil

f riction(min. elevation)

(Ibs)

230+.25(^1H^)
4.18

+.3x1260=700 Ibs.

Recuperator mean pressure (Ibs/
sq.io)

473+710

=590

Required constant counter re-
coil orifice at max .elevation

8

1.25x4.18

(K=1.25)

Required constant counter
recoil orifice at 0° elevation
(sq.in)

Vol

13.2/p'A-Wt

13.2/590x418-290

-1260X.9848
.0645 sq.in.

1.25x4.18 x3.5
12.2/4.73x4+18-700
=.0765 sq.in.

654

0.0645.0.0765

Entrance buffer area
(sq.in)

KAV / 1
"b ' TTT /

.00894A2v* 1.33x4.18x3.5
Pa «

13.2

.00894x4.18 *3.5
590

(.0705)a
v = 3.5 f tXsec(approx)

Layout entrance area of buffer, with re-
quired depth in groove. Decrease depth of groove
to zero at end of buffer du .

Deflection from free to
solid height of spiral
spring (in)

hso s 2h" * 2x0.1862 * 0.373 in.

_ _ ____ ___ ___ _

Spiral spring constant
(Ibs.per in. )

Rsc a 2280

83 " hs0 3 .373

This spiral spring will be too bulky for practical
purposes. Therefore we will let the Belleville
spring washers take care of all conditions at
different elevations and design the spiral spring
strong enough to keep the valve closed when gun
is in battery.

Spiral spring reaction
at short recoil

R; -s,<k'*h0). o

h0«.0197 (initial lift)(in.)

655

Load on Belleville at
short recoil (max. elev )
(Ibs)

Rb 3KP~Pai)a+Paia'}~Rs= (4500-473)0.747-0= 3010
! _ lbs'

Load at solid height on
Belleville washers(lbs)

Rbo= IRb = f x 301° - 4520 Ibs.

Deflection from free to

solid height of Belle-

v i lies (in)

hbo=3h« = 3x0.1405 = 0.422

Belleville spring con-
stant (Ibs. per in)

Rbo 4520

" 1070°

compression of Belleville washers.

n = no. of Belleville spring washers =

0.422 _ 6

.071

hfc=initial compression = 115
h0=valve clearance .0197

Ratio of

DB8IGH OF CAM MECHAHISM AtiD LAYOUT.

Cam Movement

Valve movement

g = 5

(taken usually at 5)

In general, assume the length of recoil
at horizontal recoil constant from 0j to #t de-
grees, (usually from 0° to 20° elevation); then,
decrease the recoil proportionally with the
elevation, that is:- 20°

656

Length of intermediate recoil
(ft)

45-30

80-0) +30

Kg Total resistance to recoil:

0.47WrVf 0.47x1260x51.5

32. 2b
48800 58700

Pg2»Total recoil pres-
sure »

k,+1260sing
1.1

- 210

b * lift corresponding to required throttling
opening (inches)

.098A«V,

0.178V,

0.178x47.4

(p-Pai>*

8.45

t

fc"

k,

.in*

K +•

p

-k

(p-

(p-

h

sin0

p*i>

».!>•

20

45-.OO

13O5O

. 342O

13430

120OO

10OOO

2390

48. 89

. 1725

25

43.75

134OO

. 4226

13930

12470

10470

2500

50.OO

. 1680

25

41. 25

142OO

.5736

14920

13370

11370

2710

52.O6

. 1624

50

37.50

15650

. 766o

1662O

14900

12900

3030

55. 50

. 1520

65

33-75

17400

.9063

18540

16640

14640

3500

59.16

. 1425

75

31.25

18750

.9659

19970

17930

15930

3800

61.64

. 1370

BO

3O.OO

19550

. 9848

20790

18690

16690

3980

63.09

.133fi

Linear notion of cam rod against elevation.
sidering spiral spring reaction negligible.

Con-

h + h8 -

(P-pai)a+paiat

0.175+h-

(p-pai)0. 747+127

10700

657

a

b

o

d

e

f

X

Vfax.of

cam rod

inches

3

b"

h cor-

h +

(P-

3 + 127

f

d-g

rected

0.175

0^747

10700

20

45^00

. 1867

. 3612

1785

1910

. 1784

. 1828

.911

25

43.75

. 1800

.3550

1965

1990

. I860

. 1690

. 845

35

41. 25

.1110

. 3460

2030

2160

. 20 20

. 1440

. 720

50

37.50

. 1600

. 3350

2300

2430

. 2270

. 1080

. 540

65

33.75

. 1500

. 3250

2610

274O

. 2560

. 0690

.345

75

31.25

. 1450

. 3200

2840

2970

. 2780

.0420

. 210

80

30.00

. 1405

.3155

2970

3100

. 29OO

.0255

. 127

105 M/M HOWITZER

75 M/M GUN (Double Charge)

MOUNTED ON SAME CARRIAGE.

Given:

d - diameter of the bore (in)

75 m/m Gun
Normal
Super

105 m/m
How.

2.953in.

4.134in

v = nuzzle velocity (ft/sec)

1500
2175

1500

w = weight of charge (Ibs)

1.401b3.
S.OOlbs.

3. 25 Ibs

u = travel of shot up bore (in)

109.50in.

BO.OOin

0m= max. angle of elevation

80°

0^= min. angle of elevation

0°

w - weight of projectile (Ibs)

151bs.

331bs.

Pjj=max .powder pressure on
base of breach (Ibs/sq. in. )

34,000

24,000

bg = length of recoil at max.
elevation

1.3ft.

2.5ft.

t>h=length of recoil at 0°
elevation

2.4ft.
3,75ft.

3.75ft.

658

WBIQRT Of GUN AND CARRIAGE.

Similar Guns

W

V

E=Muzzle
Energy

*g

E/wg

*t

w=
X* wt.

75 BB. French

16

1700

716000

1050

705

2657

39

75 mm. U.S.

16

1700

716000

750

956

3045

25

75 mm. British

16

1700

716000

995

720

2945

29

3.8 How.M.

1906

50

900

378000

432

876

2040

22.6

4.7 Gun M.

1906

60

1700

2690000

2688

1000

8068

33.6

4.7 How. M.

1908

60

900

755000

1056

716

3988

27.

6" How.M.

1908

120

900

1510000

1925

785

7582

25.7

155 m/m How.

(Sch)

95

1420

2970000

2745

1080

7600

36.5

155 m/m Gun
(Fil.)

95

2300

8400000

8795

960

25600

34.5

155 m/mHow.
(St. Cham)

95

1520

3400000

3040

1120

7700

25.3

8" How. VI

200

1300

5250000

6652

790

19100

35.0

8" How. VII

200

1525

7200000

7730

933

20050

38.7

Average E/wg of

888

E/wg=1000
E/wg=888

v 7

normal gun

super gun

howitzer

15x1500
64.4x1050
15x2175
64.4x1000

33*1500

64.4x1000

1100* 1240
*1150<( 1290

1100+30 » 1130

Wr How. 1180+30 = 1180

Average 1155#

1260#

659

Using highest % of *»r to Wt (39*)Wt - - = 2970*

397

2970 - 1155

1815f

Wr Weight of recoiling parts 1230* and 1260*
for gun and howitzer respectively are the
minimum weight that could be used on account
of stresses.

The condition being to get the minimum
weight. These values are used:

W. » 1230* gun

»r * 1260* how.

Gun
super

38.00

105 m/m

How.

51.50

7

'.,,. i. r,.tL -in-' -; "I--5" _r ^ i-ll.inji

5m/m

ormal

23.60
1

Vf max. free velocity =
wv+4700 w

15x1500+4700x1.4

1230
15x2175+4700x3

1230
33x1500+4700x3.25

1260 "
sec

Ks Resistance to recoil
at 80° elevation

wr

= 1.05[ — V2*
2g r
1

uV,1
b_ + (.096+. 0003d) — *•
2 v

1230x23.60

1500

660

•1.05U0640* 1

1.5(. 096*. 0009)1. 724

1 . 1.05x10640 fl__-

'1.05(10640* )= * o720

1.5+.167 1.662

_f 1230x38. 00 __ 1
64.4 "X1.5

1.05x27600

17600

1.645

1260x51.50

64.4 V

2 .5+ ( . 096+ . 0003x4 . 134 )

1500

1.05*51900 . 19700

2.761

h = height of axis of bore above ground. Assumed 36"

bh = max. allowable horizontal recoil » /

2g

, /23'60 x3 51", 82 ",1004"

64.4

Vr * max. velocity of constrained recoil ,9Vj(app)

21.20,34.20,46.35

A « recoil constrained energy = -£ —
1230x21.20

8580

i

1230x34.20

64.4

22400

64.4

1260x46.35

42000
64.4

661

E * recoil displacement during powder period.

, 1B n."^~

1230

12

1230x12

.15+.5X3. 109.5 2.24x16.5x109,5
2'24( } - * '275 ft

1230

1230x12

33+.5x3.25x 80 2.24x34.63x80
2.24( - ) — = - = .41

1260

12 1260x12

STABILITY

ls -

3000

.. 6.35.n.

c = constant of stability = ' « .96

stabilizing moment

efg * angle of stability 20°

d - moment arm of overturning force

htcos0+d8=l,,sin0=36xO. 9397+7. 5-81x0. 3420 »

V O

13.60 in. =1.13 ft.
bn = length of recoil at the angle of stability

¥sls+WrE cosg±/|*sIs+WrEco8g)*-4Wrcos0(WslsE -)

662

3000x6. 35+1260*

0.4lx.09397+y^Hsl8+WrEcos0)11-

2x1260x0.9397

4xI260x .9397(3000x6. 35x .41)

.96

19, 540*/119, 540) '-4740 (7800-49500)
2370

19. 540+/110,000, 000 19.540+10490

2370

2370

3.75 ft,

RBCUPgRATOR FOHGIBG8.

Approximately

= maximu!n resistance to recoil

.45 r

.45 1230

. . . nn
-- VI * - - 23.
b g 1.5 32.2

.45 1230

„
60

1.5 32.2

* 38.

2.5 32.2

* n in. resistance to recoil
0.47 ^r y, m 0.47 J[r

3.75 32.2

75 »» Gun 105 *•

Mora- Super How.

6380

16500

18700

2660

663

•

0.47 1230

3.75 32.20

0.47 1260
3.75 * 32.2

-^
&1.5

Pg = max. pull = KS approximately
Pfi = nain. pull = Kh~0.3 Wr

Kv = initial recuperator reaction*
1.3Wr(sin0_+.3cos0_;

X ill til

= 1.3xl260(. 9848+. 3x. 1836)
» 1.3x1260x1.037

r »

recuperator cylinder area
eff.area of recoil piston

A.

~ » .039V,

.039x23

.60 /—

2290-1700

10.1

Ph-1700

75 —
• or*

al

Gun

-Super

lOg KB

HOB.

6740

13100

•380

16500

18700

2290

6370

12730

1700

5.35

664

_ 19000+17150

Ph = 3560

paf final air pressure

n ' - — « . . . . : (generally) 1.5

Paji initial air pressure

e = length of air column from chart 1.25b
A. = effective area of recoil piston =

ps 19000

4500 4750

4.00 sq.in.

(Usually packing is designed to stand a pressure
of 4500 to 5000 Ibs)

PS * max. pressure corresponding to r = 3

P^ = I222?= 4750 Ib3<

4

WQ = total weight of recoil piston and rods, 30 Ibs.
T. * max. tension on the rods at 0° elev.»K<- + ffr

30
19300+13.45x24000* =27000

1260 t
= assumed fibre stress = - elastic limit

max

70000

« 35000 Ibs/sq.in.

27000
Ar - area of the recoil rod = = .772 sq.in.

35000
dr = diameter of recoil rod = 1.127

/ar = 1.127/772

+ .99in.(nake 1 in.

Ar» total area of recoil cylinder = 4. +.781 - 4.781

sq.in

Dr 3 inside dianeter of recoil cylinder -
1.127/~Tr « 1.127/4.781 = 2.46 in.

Ar = rA * 3x4 = 12 sq.in.

Da * 1.127 /TI = 1.127 /T2 = 3.9 in. diaicster

a a «

of float-
ing piston.

665

CALCULATION OF PACKING FRICTIOH.

b = O.lSin.to 0.25in.use 0.21in.

a = depth of outer silver flange of packing cup 0.14 in,

to 0.16 in., use 0.14 in.

a1 = depth of outer silver flange 0.18" to 22", '0.18"
c t = constant spring comp.of total packing friction

a1

ct = n(Dr+dr)[.05b+.09(a+ —-)]0.15 Pmax

45

= n(2.46+.?81;r.05x.2H-.09C.14+.09)).15 Pmax
» 10.2(.0105+.0207).15Pmax

=10. 2x.0312x. 15x4750
c1 = 226

a1

C_ = n(Dn+d_) [ .05b+.09(a+ — ) ] . 73

JT I I Q

= n(2.46+.781)[.05x.21+.09(.14+.09)].73
= 10. 2x. 0312". 73
C = .232

2

Rp = total recoil packing friction - ct+c p(p=

Ibs/sq.in)

226*. 232 x 4750 1326 Ibs.

FLOATING PISTON FRICTIOH

*.i>3
cls cast(spring constant) of floating piston =

at

1« 12 ^Do I •05b^*»09(.£L +~"~*) j p o f = G P a ^

3 ' d I o 1

= 1.12 x n x 3.9[.05x.21+.09(.14+.09)]paf= GPaf
= 1.12 x n x 3.8 x .Q312Paf

= ,428Pa£=GPaf(in Ibs) Paf = final air pressure

^ = pressure constant for fluid pressure corap.
of total packing friction.

-

= 1.46nDa[ .05b + J

= 1.46nxD x. 0312=1. 46 x n x 3.9 x .0312

•

= .558

666

D2SI6N OF RBCUPSRATOR

75m/m Gun 105m/m
Normal Super How.
6720 17600 19700

b_+(.096+. 0003d)
Kh = min. resistance to recoil

B V! i

uV

b + (.096+. 0003d) — -
v

1230x23.60
64.4

1500

10640

bh+.167

- «
1230x38

(bh=2.4 ft. =29 in.)
1

27600
- -
3.925

1260x51.5

4140

64.4 3.75 +.267

^ — t

1260x51.5

-

64.4x4.017

max

226+ .232

6720

7030 Ibs.

12,900 Ibs,

616 1236 1366

1 * distance between clip reactions

60in.

- 616

- 616

7044

1.0388
17600*1210

- 1236

16900

1.0388

19700-»-1240
1.0388

- 1366.

Pb=pull at horizontal elevation in Ibs.

3560

667

r1 = distance from center of gravity of

recoiling parts to mean friction line. 6. Sin.

n » coefficient of guide friction (.1 to .2) .15

d^ * distance from center of gravity of recoil-
ing parts to axis of piston rod 7. Sin.

PULL OH THB ROD

75 m/m Gun
Normal Super

Pg=pull at max. elevation
K8+Wrsin0

2nd"L
1-2 nr

6720+1230X.9848

— ___^_^_— _

2x.l5x7.5
60. 2x. 15x6. 5

6720+123QX.9848
1.+.0388

7930

18800

2ndK
l-2nr

- C

668

•

p ,( - D -- 226). 945
1.0388

(3560x1. 0385+226) 1.0388=Kh=4140
7030

1.0388

- 226)x.945 6180

12900

Ph=( -- 226)*. 94.5 11550

1.0388

R * excess recuperator battery reaction

constant 1.2

(1.1 to 1.3)

Kv* recuperator reaction in battery at
max. elevation

R(WriinBm*c ) 1.2(1260«. 9848*226)
KT. ,

2w "1 C „ *7 C OOO

1-R( +_^j l 1>s^ *.1&*7.5 ^ .233

1+2nr A 60+2x.l5x6.5 4

1.2x 1466 1760 1760

___^______^__^___ — _______ — ____^^_

1-1. 2(. 0363+058 1-.1128 .8872

Ky a 1980

Vr - .92Vf=. 92x23. 6=21. 7=Vr; 35.00 = 47.40

_ _ o enc tr / S _ o coc^oi " / 188QQ

AC

r - — - 2.625 Vr / '-* » 2.625x21,

A P (Ph-Kv) 4750(3560-1980)

57

4750x1580
r « 2.85

669

r.la . 2.625 Vr 168°° . 81.6

4750(6180-1980) 4750*4200

= 2.79

af

- 1.5 = B

1 - l.Sb

ps 18800

A - » - - 3.96

4750 4750

Aa = rA = 2.85x3.96=11.30 sq.in.»Aa

Pjj=nin. pull on the rod

Vr = velocity of recoil corresponding

R » 1.295

w? =U0373 rA>2

h "v

r .2435

175x. 00139 (Ph-Kv)

Pp = pressure the packing should withstand

P*V«
6.9 — S-1-

Pp(Pn-Ky)

r » 2.625 V.

j = length of air coluran in terns of max. recoil
1. 1.3b

670

j - 1.3

• *( r'1 ) ' from the chart 1.5
r.J-1

Ky 1980
paj = initial air pressure (Ibs/sq.in) = -— * — —

A o » <7w

Pai * 500 Ibs/sq.in. (approx.)

Pif * * Pai = !-5 x 500 s 75° x paf approx.

VQ * initial air volume = Aa x ia = 11.30 x 45 »

662 cu.in. = V&

- 1

INITIAL AMD FINAL AIR PRBSSURB AMD

AIR VOLUME.

50°

Rt » floating piston friction initial

3 Ci+C« Pai = -428xPaf +.558^500

= .428x700+. 558x500

= 321+279

Bt « 500

P » Pli + -1 - 2 ai = 500 = - » 500+44.2
ai Aa 11.30

Pai - 544 Ibs/sq.in. Pai = 540 Ibs/sq.in.

paf = * Pai«l. 5x544=816 Paf = 810 Ibs/sq.in.
V0 =662 sq.in.

Vf = final air volume = VQ-Abh=662-3. 96x45= 662 -178
Vf = 484 cu.in.

671

Pa » average drop of pressure across floating pis-
ton = Ct+Ca(Pal+Paf)0.5

540+810 540+810
* .428 x — + ,558x

= (.428+. 668)675=. 986x675
Pa = 665 Ibs.

W- = 30 Ibs.

V

TL * tension on the rod =Ks+» ^

30 r
= 188000+126Qx. 9845+13. 45x24000*

1260
= 18800+1240+7700

TL » 27750 Ibs.

Ffflax = assume fibre stress 1/2 elastic limit
= 32500 Ibs/sq.in = Fm&x

27750
A_ » area of the recoil rod = • » .853 sq.in.=a_

32500

dr « diameter of recoil rod 1.127/.853 = 1.04in» dr

A'* area of the recoil cylinder = 3.96+1.04 =
500 sq.in.= A'

Dr * 1.127/5

2.52in.=D

r

D =1.127/11.30 =3.78K=Da diameter of air cylinder
W* - 30 Ibs.

STRENGTH OF CYLINDERS;

Test pressure = 2 x service pressure
Pt = max. allowaole fibre stress for cylinders =

3/8 elastic limit = 3/8 x 60,000 = 22500 Ibs/

Raft » min. outside diameter of recuperator

lao

1.S9 A

22500+810

Pt-P... 22800+810 21700

U <a 1

= 1.958
Dao = 3.92 in.

672

ro

ro

/
/

22500+4750

82500-47S0

•1.561

• 3.122in.

W * «in. width between adjacent cylinders

PaaxDr*Pafxpa
1.5Pt

11950+3060

33750
.445

4750x2.52+810x3.78
1.5x22500

.445in.

MAXIMUM AND MINIMUM THROTTLIH9 AREA.

75 ID/ID Gun 105m/m

Normal Super How.

maximum throttling area

(at 0| elev. )

.098x3.96

673
.7727,

.772x21.7 16.72

* 1 — » rr-rr .422

(3560-1980)* 39.75

.772x35. 27

(6180-1980) 64-81

.772x47.4 36.60

' 1 = .374

(11550-1980)*

tfh » .422 sq.in.

fg * minimum throttling area (at 80°
elev. )

•772 vr 16.72 16.72

(7044-1980)* 71.16

.237 sq.in.
27 27

* = IS27F -221 s*iD'

(16900-1980 )s

36.6 36.60

282 sq.in

129.7 (18800-1980)

.221*. 282
a ________ = ^25 sq.in. = ws

2

LAYOUT 0? RECUPERATOR FORGIHQ

If = overall length of forging = 1.5 * bQ » 1.5 *45 *

67.5 in.
DP - diameter of recoil cylinder 2.52 in.

674

Da = diameter of air cylinder 3.78in,

la = length of air column 58. 5 in.

Wr * area of connecting channel between re-
coil and recuperator cylinder.

Wa = 3.5 to 4.3 Wh

= 4x.422 =
Wfl = 1.70 sq.in. da = 1.468 in.

D1 = maximum depth of recuperator cylinder
below recoil cylinder =

= 2.52 -.734
D' = 1.786in.'

Wc = const, channel area from regulator

valve = 4.3 Wh = 4.3x.422
Wc = 1.814

ho = depth of const, channel area, Wc in inches.
h0 = 0.2Da = .2x3.78 = .756in.=b0

a = extreme area of regulator valve =
W? 73.5x.~422

73.5

D| (3.75)*

a * .933 sq.in. y

da = diameter of entrance channel - 9.675 -— =

422
0.675 = 1.09in.-^da

3.75

DESIGH OP REGULATOR.

a = area at base of regulator valve = 73.5 — — =

73.5x.422
.93 sq.in = a

(3.75)2

h 422

d. = dianeter of a. 9.875 — = 9.675 = 1.09=da

Da 3.75 i

da * diameter of upper and lower valve sten = 0.6da
.6x1.09 =0.655 = .d_

675

at = cross section of stem = 0.36a=.36 *.93 =

.335sq.in. -a.

^, = diameter of inside foot of valve stem =

1 .5 to .6 da
ai

= .55*. 655 = .36in. = da

MM.

c1 = length of flap at base of scale = 1.571 da -.4725

1.571 x 1.09 -.4725 - 1.237 in.
c = effective circumference = 1.571 * 1.09 + .4725

2.184in.=c
Rt = reaction on tbe valve at short recoil

3 + 500*. 335

= 3950+170
Rt = 4120

SPIRAL SPRIHG DESIGN FOR REGULATOR VALVE.

fs = maximum torsicnal fiber stress 120,000 Ibs/sq.in.
U = torsional modulus 10,500,000 Ibs/sq.in.

ds = let dianeter of the spiral be 4 diameter of
the wire

8PD 32P 10.16P

*§ = r> d* - — --T—

676

£P" 496 ~
« 3.19 A = 3.19( )° =* 3.19 x .0643

ds

f 120000

.205in.

D8 * diameter of helical spring 4 x d » D_ » .82in.

deflection per coil
nfsDI

Gdg ia500.000x.205
f = .118

496

- = 4200 Ibs. per inch of deflection required

.118

248

- = 1162 spring const.

.213

4200

- » 3.61 effective coils

1162

n = no. of coils = 3.61 + 1 = 4.61 use 4.5 coils

Ph
Rs = load on spiral springs at 0° elev. = -— - Pa

*l

o c c r\

= ( -- 500). 93 = 400 x .93
3.96

Fs = 372. Ibs.

Wp ^422

h" lift of valve at long recoil = — - - = .193

c 2184

h" * .213 inches Valve seat clearance = .02

"W 25 .213in>

h" - lift at short recoil = — » « .1144

c 2 . 184
h" =.1144Jn.

Rsc = load at solid height of apiral spring -j- x 373 »

496 Ibs.

hgc = deflection from free to solid height of
spiral = 2h" = 2x.213

hsc = .426in.

677

g

Sg* spiral spring const. — — = » 1162 # * S_

h__ 426 ~_

O C *-^ -.^ii^^MiP-^Bta-

g * spiral spring reaction at sbort recoil

Ss(h" + .02)»1140(.1144 + .02)
R' - 153 Ibs.

INITIAL
DEFLECTION

WORKING

*«

N
<0

DEFLECTION

»

SOLID
HEIGHT

.1OG9

-.2.13

« — .»O65

* load on Belleville at sbort recoil

18800
( - .500). 93-. 355x500

3.96
(4750-500). 93+177. 5

3770 #

at solid beigbt of Belleville washers

' I Rb = ! x 3770
Rbo = 5650*

hu0 * deflection of Belleville from free to solid

height 3h'=3x.H44
hbo « .343*

678

Sb * Belleville spring const
Sb - 16080

6512
.343

16080

*o

h-

Si

CU

10

-.ie — •

*-.1064-

-.646*

DESIGN OF CAM MECHAHISM AND LAYOUT.

g = ratio of can movement to valve movenent usually 5

X = distance valve should lift to engage Bellevilles

S^ = working deflection

hg - initial corap.of spiral spring

h0 = clearance of valve

h = lift of valve

hjj = initial compression of the Bellevilles.

X = — {ss(bs+h0*h)+Sb(bb-i-h0+h)-[(P-Pai)a+Paial]['

Rs=S8hs+Ss(h+li0) (Steins of two springs are in con-
tact)

(P-Pai). 93+500*. 355

1162*. 1066+11. 62 (h+. 02)
+16080*. 1144*16080 (h+. 02)

679

124 +1162h+23+1840+16080h+322=(p-pal). 93 + 177. 2
17242h+2132= (p-pai ) .93

18500
.098AVr

<P-Pal)«

.098x3.96*7,

2.184(P-Pai)2

.178V,

ai

(1)
. 098x3. 96*V

ch»2.184 h

~ C !

.178

a

b

o

b

V

2290

Pai>¥

2660

360

.0194

51.58

1.00

5. 60

3000

7oo

.0378

54.77

2.07

11.62

3250

950

. 0512

57.01

2.92

16. 40

3500

1200

.0648

59.16

3.83

21.50

3750

1450

. 0781

61. 24

4. 78

26. 90

4000

i7oo

.0917

63.25

5. 8O

32. 60

4250

1950

. 1050

65. 19

6. 85

38.50

45OO

2200

. 1188

67.08

7.95

44. 6O

Normal

LENGTH Of RECOIL

80° Elevation

Vr = 21.70 corresponding (P~Pai) from

curve 3500
Ps = (3500+. 500)3. 96=4000x3. 96-15820=Ps

680

-226+ . 232x4000-226+930
RBajt-1160 Ibt.

Kg+»rsin0
Ps * 2nd8 R««

l-2nr

Ka+1230x.9848

15820 - — 1160

1+.0388

16450-K.+1210-1200 N y.
Kg « 16440 - 1.05 [ '

2fi uVf'

b.+ (.096+. 0003d)

v

,,. 1.05x10640 11200
16440 » -

b+.167 b+.167

16440b+2750-11200

8450
b » » .513

16440
b - 6.17ia.

Super

Vr » 35 corresponding (p~Pai )*4100 Ibs
Pg - (4100-500)x3.96=4600x3,96
P, - 182001

K8+Wrsin0

Ls

2nd,
1 +

l-2nr

226+. 232x4600-226+106

Rmax ' 130°

Ka+1210

P. » 18200 = - - 1300
1.0388

18880»KS+1210-1350

681

W_V|
K, « 18020 lb«. « 1.05 I— -

1.05x27600

X -^— — ^— — —

b+.145
19020b+2760»29000

b « 16.55in.
Howitzer

Vf»47.30 ft/s.c. (P-Pai)-4620 Ibt
P3 = 5120 " 3.96 = 20250 Ibs. «
Ks+Wrsin0

2ndb
1- 2nr

~ Rmax

Raax s Ct't'CtP * 226+. 232*5120 » 226+1190

Rmax ' 142°

IT8-1240
Pg » 20250 - -- 1420 -2100OK.-1240-1470

1.0388

Kg»21230 lbs.=

21230 = i-

b+.267
21230b +5630=54500

b'- -2.3 feet
2123

b"»27.6 inches.

bs = (80-0)+30
60

1(80-0) +30

682

1(80-0) +30

4

1.05x51900
b+.267

655000

54500

- +.267
12

655000
b"+3.21

b"+3.21
C

—

3.96

Ks+Wrsin0

2nd

- C.

1-2 nr
Kg+1260 sintf

- 226

Pc =

K_ + 1260sin£)

9

1.098
,178 Vr

1.0388
- 212.

.178x47.30

8.43

.178

0°

20

25

35

50

65

75

80

b "

45

43.75

41. 25

37.5

33.75

31.25

30

*,

13600

13950

14710

16050

17720

19000

19700

s inflf

. 3420

. 4226

. 5736

. 7660

.9063

.9659

. 9848

K.*

»r

• in^

14030

14480

15430

1702O

18860

20220

20940

Km*r »int

r

12800

15200

14100

15500

17200

18900

19100

1.098

P.

12600

13000

13900

15300

17000

18700

18900

(P.-P.i)

1060O

11000

11900

13300

15000

i67oo

16900

P^-KV
3.96

2680

2*780

3000

3360

3790

4220

4270

683

(p-pai)¥

51.77

52. 73

54.77

57.97

61.56

64.96

65.35

h

. 1628

. 1596

. 1536

. 1452

. 1363

. 1297

. 1286

1162(.1065+.02+h)-H6080(.1144 + .

.93+500X.355]
1

Ps-1980
(1162h+147+16080b+2150l ( ) .93+178]

16080 3.96

— (17240h+2300-.235Ps-290)=— - — (17240h-235Ps

16080 16080

+2010
1.072h-(.00001463P8-.1251>

16

h

1.072h

PS

1.463
105 8

b

X

5X

20°

.1628

.1749

12600

.1842

.0591

.1158

.580

85°

.596

.1715

13000

.1900

.0649

.1060

.530

35°

.1536

.1648

13900

.2034

.0783

.0870

.435

50°

.1452

.1559

15300

.2390

.1139

.0420

.210

65°

.1368

.1469

17000

.2490

.1264

.0205

.103

75°

.1297

.1392

18700

.2735

.1484

.0008

.004

80°

.1286

.1380

18900

.2765

.1514

.0000

.000

Counter Recoil
Buffer, constant orifice and port Areas.

R' = packing friction at end of counter recoil
Ct = 226 Ibs.

d » recoil length during buffer action T""

684

0.238Wr

.238x1260 300 300

z — — — ^— — — — = i a —

15^1260+226 189+226 415
db - .723 ft « 8.7 in.

A db
d^ « length of Duffer rod * 1.2 x dfax — , 1.3 —

Aa r

1.2 x .723

, » .3 ft » db - 3.6 in.

2.9
Length of buffer chamber * 1.2 to 1.3 db'

dg » 4.5 in.

allowable counter recoil velocity

2.5 to 3.5 ft/sec.

total counter recoil friction - aax
elev.

Wr(»in 0t0.3 cos 0a
Ct + CtE - - ]+0.3Wr

3.96
.1736

..3X1260

« 226 +.232(.2615 )+66

Rt " 290 Ibs. Max. elevation

, +C ( - ) +.3HP« 226+96+378

A

'£ * 700 Ibs. Win. elevation.

pai+pap

max. recuperator pressure = P^

500+750 2

— » 625

685

c'recoil orifice at 80° elevation

KAX

• (K»1.25)

1.25«3.96 x 2.5 24.6 24.6

— — — ^^— — -— ^— — — —^^— ^^^— ^^^— -— = i 3 — — —

13 . 2/625x3 . 96-290-1260x9848 13.2/670 411
.06 sq.in

0i

34.5 34.5

'o " =

13.2/PaA-R{ 13.2/2470-700 13.2x38.35

34.5
506

* .0682 sq.in.

— = .0641

KAV

entrance buffer area =

13.2 0.00894A«V«

1.33x3.96x3.5

13.2 .00894x3.96 x3.5

625 -

(.0641)2

/ 1 1.395

1.395 / * » .097 sq.in.

625-418

Lay out entrance area of buffer, with required
depth of groove, decrease depth of groove to
zero at end of buffer dJ..

CHAPTER X.

RAILWAY GUN CARRIAGES.

TYPES OF MOUNTS. For coast defense or other
use of heavy artillery, it has
been accepted that mobility is
of great importance.

Materiel in permanent emplacements is more readily
subjected to attack. Further with long coast lines
it is impracticable to supply enough permanent
batteries for adequate protection. By introducing
heavy mobile artillery ire increase the protection
and develop the advantage of concentrated fire at
any one point when needed.

Railway artillery meets the demand for
mobility in a very satisfactory degree. Very
heavy weights, as occur with large caliber guns
and their corresponding mounts, are most readily
transported by rail. Hence there has been a
tendency of development along two lines; first,
a mobile railway carriage that is entirely self
contained and fired directly from the rail and
(2) a mobile mount, transported by rail but set up
on a semi-fixed emplacement. For extreme Mobility
the first is most useful, wherein for coast defense
work the second plan offers many advantages.

Railway carriages have been developed along
the following lines. In their methods of firing.

(1) Sliding carriage type with no
recoil mechanism , the carriage
merely sliding back during the re-
coil along special constructed rails
or guides, trucks being disengaged.

(2) Railway carriages with a recoil
system, the whole carriage in ad-
dition recoiling on special ways on

887

688

rails, the trucks being disengaged,
or the trucks being engaged and
the secondary recoil being direct-
ly along the rails.

(3) Fixed or platform mounts. With
light railway artillery, the car
is held stationary by suitable out-
riggers and we have usually a bar-
bette type of mount, mounted on
the car. With heavier types, the
girder which supports the tipping
parts is placed on a large pintle
bearing with sometimes additional
support at the tail of the girder
with a circular way or track for
all round or sufficient traverse.
In this latter type the trucks
must be disengaged and the main
girder run on to the permament
emplacement.

The sliding carriage type (1), was developed
successfully in France and was considered satisfactory
during the late war. This mount, however, is sub-
jected to the direct firing stresses with consequent
requirements for a very heavy girder and trunnion
support. It has on the other hand the advantage
of doing awaj with a recoil system. At best, how-
ever, it can be regarded merely as an emergency
type of carriage that might be developed under
great stress of war pressure and not suitable for

use against moving targets.

In railway carriages of type (2), we have

virtually a double recoil systew. However, since
the recoil is designed for stationary service
as well, or for the condition at max. elevation
where the secondary recoil is small, the maximum
reactions at the beginning of the recoil are the
same as in a. stationary mount, with a single
constant recoil. When the trucks are disengaged
a specially built track must be laid, and the

689

girder slides back on friction shoes, which are
lowered to engage with the track. Mounts of this
type are illustrated in our 14" railway mount
ME. When the trucks are not disengaged and the
secondary recoil takes place on the track, the
bearing reactions of the truck wheels must be
suitably designed to sustain the additional firing
load and the trucks must be suitable braked to
resist the secondary recoil, and bring the mount
to rest after the firing. When a built up track,
trucks disengaged, is used the successive firings
must necessarily take place along the tangent of
the track, whereas firing directly from the rails,
permits the use of a curved or Y track, and con-
siderable traversing is thus possible by the
firing taking place at different points on the
curved track. With railway carriages of type (2)
very little traversing is possible on the mount
itself and therefore the track must be laid very
closely in the direction of firing. In railway
carriages of type (2), we are greatly limited by
road clearance. For clearance, the trunnions
must therefore be in the traveling position in
a low position. On firing however, at maximum
elevation, the recoil becomes limited. To pro-
vide for a suitable recoil at maximum elevation
the trunnions are raised and a balancing gear
throwing the trunnions to the rear may also be
introduced.

With fixed or platform mounts of type (3),
the special features are the methods of
erection on to a serai permanent emplacement
and the disengagement from the traveling con-
dition of the mount . We may have a center turn
table which serves for the pintle in traversing

and the tail of the girder is supported by a suit-
able circular guide which balances the overturn-
ing moment and thus releases the otherwise bend-
ing or overturning moment on the pintle bearing.

690

With this type of mount large traversing is complete-
ly possible.

SPECIAL PIATURS3 IN THB DESIGN.

Recoil System:

(1) The recoil should be simple and
rugged.

(2) A constant recoil or approximate-
ly constant for all elevations

should be used.

(3) A constant resistance to recoil
is satisfactory since questions of
stability are not usually of prime
consideration, and the recoil is
thus simplified.

(4) The counter recoil should be
sirople, an ordinary spear buffer
being usually satisfactory although
other control may be sometimes
necessary. Bere again counter re-
coil stability is no longer a con-
sideration and high velocities of
counter recoil are not objectionable
provided there is no shock at end

of counter recoil.

(5) With very large guns used at high
elevations, high pressure pneumatic
recuperator systems should be used
in place of spring columns, since
the weight and bulk of springs be-
come excessive.

(6) Sleeve guides for the gun have
been found most suitable and tne
various pulls should be so far as
possible symmetrically spaced about
the axis of the bore, thus reducing
tne bearing reactions in the sleeve
and making it also possible to keep

691

the center of gravity of the recoil-
ing parts close to the axis of the
bore.

Tipping parts:

(1) The cradle should be of the
sleeve type thus reducing the
bearing pressures over guides
and clips.

(2) The recoil and recuperator can
be strapped on with suitable
shoulders for bearing surface to
take up the recoil load from the
cyli nders .

(3) The trunnions should be located
near line through the center of
gravity of the recoiling parts and
parallel to the axis of tue bore.
This reduces the elevating re-
action during the pure recoil to
merely that due to the moment ef-
fect of the recoiling parts out of
battery.

(4) Great effort should be made to

locate the center of gravity of the
recoiling parts as near the axis
of the bore as possible either by
symmetrically distributing the re-
coil rods and attachments or if
necessary introducing counter
balancing1 weights. Thus the
whipping action during the powder
period is reduced with a correspond-
ing reduction in the elevating arc
reaction during the powder period.

(5) With high angle fire ^uns or
howitzers, the trunnions may be
thrown to the rear, and balancing

692

gear introduced, thus making long
recoil possible. Another plan for
accomplishing the same results is
to raise the trunnions before fir-
ing.

(6) The trunnion bearings should be
supported on springs during travel-
ing, though compressed so we have
solid contact during firing.

(7) To reduce the friction during

the elevating process, ball or roller
bearings should be introduced in
the trunnion bearings, or in an
inner trunnion should be introduced
of smaller radius than the main
trunnion for reducing friction on
rotating the tipping parts.

LIMITATIONS IN With heavy artillery mounts,
BRAKE LAYOUT, either railway or lor permament

or mobile emplacements, counter re-
coil stability is not a consideration.
On the other hand we are limited to
a maximum allowable buffer pressure in the counter
recoil. With counter recoil systems which come
into action towards the end of counter recoil,
practically the entire potential energy of the
recuperator most be dissipated by the buffer over
a relatively short displacement. Now since the
potential energy of the recuperator is a con-
siderable fraction of the energy of recoil, we
see that the buffer reaction is of a magnitude
comparative with the brake resistance during the
recoil. Further the effective area of the c'recoil
buffer, due to constructive limitations, is necessarily
considerably smaller than the effective area of the
recoil brake. Hence the buffer pressures with a
short c'recoil buffer, become very great. This is
especially pronounced with a short buffer and high

693

angle fire gun where the unbalanced recuperator
energy is necessarily great, when the gun c'recoils
at lovi elevations. As to the limiting allowable
buffer pressures, no hard and fast rule can be
made, but it is certain that the buffer pressures
in many of our mounts are rather too high for light
construction, requiring heavy and strong buffer
chambers .

With recoil brakes having a continuous rod
extending through both ends of the cylinder, the
effective area of the buffer must be necessarily
very snail and the stroke of the buffer short due
to the fact that during the recoil it is important
that the void displacement be not too great.
Hence this type of brake with continuous rod and
enlargement for c 'recoil buffer ram, has inherent-
ly excessive buffer pressures. It is very important
with such mounts to maintain a minimum recuperator
energy, that is to use the minimum recuperator re-
action combined with a low ratio of compression,
consistent with proper c'rscoil at maxiuum elevation,

To reduce the -buffer pressure, the c'rscoil
regulator should be effective throughout the recoil,
and thd effective area of the buffer should be as
large as possible. This actually has been obtained
constructively in our 16 inch railway mount, the
buffer area being equal to that of the recoil brake
and c 'recoil regulation taking place throughout the
rscoil. The buffer pressures are therefore compar-
able with the brake pressures during recoil.
DESIGN LAYOUT OF Assuming a preliminary layout
RECOIL SYSTEMS. has been made, the weight and the
ballistics of the gun given, we
may estimate from previous mounts,
the probable weight of the recoil-
ing parts and tipping parts.

Therefore, we will assume the following data
given or estimated from previous mounts:

694

Wr * neigh! of recoiling parts (estimated) (Ibs)

d * diameter of bore (in)

v » nuzzle velocity (ft/»ec)

w * weight of projectile (Ibs)

• » weight of charge (Ibs)

pbm 3 maximum powder pressure (Ibs/sq.in)

b = mean length of recoil (ft)

0m » maximum angle of elevation

0£ * minimum angle of elevation

u « travel up the bore of the projectile (ft)

Calculation of E and T:

From the principle of Interior Ballistics,
we have, R

PU, = - d" pb|n = max. total pressure on
breech (Ibs)

wv1
Pe » — - » average force on projectile

(Ibs) _

e - u[(~ ^ - 1)± /<l - |Z -i)« - 1 ] . twice the
P« P« travel of

projectile to
max. pressure in

27 . u

Pftw * r— e . P_ * total pressure on breech
4 (e+u)3 . .

when shot leaves muzzle.

wv + 4700ii

' » max. velocity of free recoil (ft/sec)

v

» velocity of free recoil when shot

"r

T0 • - - » time of travel of shot to muzzle (sec)

2(Vf-Vfo) Wr

Tt * - -T • time of free expansion of gears
"ob t .
(sec)

695

T » t0+t = total powder period (sec)
*o

.w+0.5 w

xfo * ' - JJ -- 'u * f ree displacement when shot
leaves muzzle (ft)

+ vfo(T~to)* free displacement
*r of recoil during

free expansion
of gas (ft)
X*0+X*»0 * free recoii during powder period, ft.

Resistance to Recoil

Knowing & and T we may immediately calculate
the total resistance to recoil for any elevation,

from the formula:- i

I "r7!
K , '• • (Ibs)

b-B+VfT

With spear buffers effective daring the latter
part of counter recoil, in order to reduce the buffer
pressure (Ibs/sq.in) the effective area of the
spear buffer is made greater than the area of the
recoil rod. Now due to the relatively snail area
of the c'recoil throttling areas, the sudden with-
drawal of the plunger of the c'recoil buffer on
firing, prevents a ready flow of oil into the space
vacated by it. Hence we would have a very great
resistance set. up unless a by-pass or void is in-
troduced. Due to difficulty in obtaining a suf-
ficiently large by-pass together with additional
constructive difficulties, it is customary to
partially fill the recoil brake cylinder leaving
a void in the cylinder.

To calculate the void displacement, fig. (1)
let

A « effective area of recoil piston (sq.ft.)

A1 ^effective area of recoil piston on c'recoil
plunger side (sq.ft)

696

697

db = length of buffer or plunger (ft)

S = length of void displacement (ft)

tg * time of recoil through the void (sec)

then, we have A(db-S)-A'd|j * 0

hence U~A ' )db

S = - (ft)

The resistance to recoil, with a void, becomes
7 mrv!
b-E+Vf (T-ts)

To compute ts we proceed as follows:

(1) If the void displacement is less
than, (w+ ![ )u

s - at)

then u1 =

wrs

(ft)

t3 = - (2.3 log — + — + 2 ) (sec)
a e e

(e+u)v

e is obtained from the previous inertia ballistic
calculations.

(2) When the void displacement is
given byL

then

where

(o -

-tn)

6mr(Vf-Vfo)

wv+4700 w

698

3 U

«" V

Since the above expression is a cubic equation
in tg, we may more conveniently solve it by sub-
stituting trial values for ts until we obtain the
approx. value of s.

In an approximate design if we assume no void,
K may be calculated immediately without the com-
putation of E and T, from the formula:

v * * m VS * W4700+WV

K » i mrvf » —xhere Vf »— — —

[b+(. 096*. 0003d) — L ] W_

v

d * diam.of bore (in)

v » muzzle velocity of projectile (ft/sec)

b » length of recoil (ft)

u » travel of shot up bore (in)

w = weight of shell

w * weight of charge

or and Wr - mass and weight of recoiling1 parts.

Estimation of Pullsr-Recuperator and Brake.

Cylinder preliminary layout.
If

Rg = guide friction (Ibs)

0m * max. angle of elevation

0£ * initial angle of elevation

Rp * total packing friction (Ibs)

B = total braking resistance (Ibs)

Pb » brake cylinder pull (Ibs)

Fvi * initial recuperator reaction (Ibs)

m * ratio of compression (assumed from 1.3

to 1.7)

Fyf * m Fvi = final recuperator reaction (Ibs)
1 = length of cradle and gun sleeve (in)
eb » distance from center of gravity of re-
coiling parts to center of pulls

699

n * coefficient of guide friction » 0.15 ap-

prox.

x1ana x2 = coordinates of front and rear clip
reactions from center of gravity
of recoiling parts (in)

A * effective area of recoil brake piston
Ay * effective area of recuperator piston
ar * area of recoil brake piston rod
av s area of recuperator piston rod

Then K a B+Rp+Rg-Wrsin0. As a first approximation,
we will neglect R and assume, R- » n Wr cos 0,
then B = K+Wr(sin0-n cos 00 (Ibs). For the initial
recuperator reaction, Fyj = 1.3 Wr(sin J0+n cos 0)
(Ibs) and since B = Ph+Fvi» tn® total braking (Ibs)
ire have for the initial hydraulic pull,
Ph « K-Wr(0.3 sin0+2.3 n cos 0) (Ibs). In a
preliminary design, the following are working
pressures, consistent with the packings:-
Ph max * 350° to 4500 Ibs/sq.in. brake cylinder-
pvi * 1000 to 1500 Ibs/sq.in. recuperator cylinder
Further let fm = max. allowable fibre stress in
the various piston rods. Hence for the recoil
brake, we have, for "n" cylinders

1.2 Ph Ph
Ar » - ; A = - (sq.in)

n fm npn max

(the factor 1.2 is to allow for the acceleration
of the rod during the powder period), and the
diam. of a recoil cylinder, becomes,

(in) and the diam. of a brake

0.785
The recuperator dimension, for "n" cylinders

becomes, p^f Fy.

a = '

where Fvf = roFvi - 1.5 to 1.7 Fvi

700

The dia«. of a recuperator cylinder, becomes,

and the diam. of the recuperator
0.7854 roa

From these dimensions a preliminary layout of the
brake and recuperator cylinders may be made, and
the positions of the center lines of the various
pulls located with respect to the axis of the
bord or center of gravity of the recoiling parts.
If now,

eh = distance from center of gravity of re-
coiling parts to line up action of hy-
draulic brake pull (in)

e? = distance from center of gravity of re-
coiling parts to line of action of re-
cuperator reaction (in)

e^ - distance from center of gravity of re-
coiling parts to line of action of
resultant pull (in)
then Fyi ey+Pneh

Cw = (in) where B = F . + P

b vi b

Calculation of packing friction.

To estimate the packing friction, we must
assume the diameters of cylinders and rods, as
approximated from the previous calculations:
then

Rp = 2 .05 itd w Pmax where d = diam. of the
various rods and cylinders

Wp = corresponding width of the packing
Pmax = (Dax' pressure in the various cylinders
The component packing frictions for the re-
cuperator and brake cylinders, consist of the
stuffing box and piston frictions respectively.

For the brake cylinder,
Rph = I .05 n(drWr+D *d)ph max.

701

where

dr = dian. of brake rod
D = iiam. of brake cylinder
tfr * width of stuffing box packing
Wjj * width of piston packing
For the recuperator cylinder R = 2 .05 « (dywy

+Dvwv>Pv max.
where dv = diam. of recuperator rod

Dy = diam. of recuperator cylinder
Wy = width of stuffing box packing
Wy = width of piston packing

then Rp = Z Rph + ZRpv = total packing friction
If Pn = the total hydraulic reaction

P£ = the total tension or poll in the brake

rods

FV = the total recuperator reaction
.Fy = the total tension or pull in the re-
cuperator rods
then Pn=Pn+2Rph ; Fv=FV2Rpv

Guide Friction

We may now estimate, more exactly, the guide
friction. We have two cases,

(1) When the resultant pulls are
symmetrically balanced around the
axis of the bore

(2) When the resultant pull is off
set from the axis of the bore.

In (1) we have simply R- = n!Tr cos Gf (Ibs)
In (2) we have

2n(B+R )eb+nWrcos0(xi-xa)

R« = (Ibs)

1+2 n eb

where n = 0.15

xt and xa are the front and rear clip reaction
coordinates with respect to the center of gravity
of the recoiling parts.

702

1 » distance between clip reactions and length

of sleeve in cradle,
e^ * distance down from bore to resultant line of

action of mean total pull (B+Rp). In general,
however, we may neglect R_ as small compared with
8, and 2 n eb as small compared with 1, then,

2n(K+Wrsin0)eb+nW_cos0(x -x )

R = x ' (Ibs)

« 1

The term n Wr cos 0(xt-x8) is usually small com-
pared with 2n(K+Wrsin0)e^ and further very often
we may assume xt = xz approx., hence,

2n(K+«_ sin0)eb
Rg - (Ibs)

which is usually sufficiently accurate for ordinary

calculations .

It is to be particularly noticed, that when

the pulls are offset from the axis of the bore,
the guide friction increases on elevating which
is exactly opposite to the condition of sym-
metrically and balance pulls about the axis of
the bore, when Rg = nWr cos 0.

[nitial Recuperator reaction,

The required initial recuperator reaction

is given by the following formula:

n cos 0 (x. -x ) ,

1+2 n er

2ev n
1 -

1+2 ner
• here Rpy = 2 .05 * (dv«v+DtfWv )pvi

= assumed initial recuperator pressure
n cos 0m )«r

703

Ay » assumed effective area of recuperator

piston

d? * dia>. of recuperator rod (in)
Dv = diam. of recuperator cylinder (in)
wy = width of stuffing box packing (in)
Wv = width of piston packing (in)
1 * length of sleeve or distance between

guide reactions (in)

ev = distance from center of gravity of re-
coiling parts to resultant line of
action of Fv

er = mean distance from center of gravity of
recoiling parts to guides ( = 0, for
sleeve cradles)

x± and xa = coordinates of front and rear

clip reactions from center of
gravity of recoiling parts in
battery (in)

n = coefficient of guide friction (=0.15)
0m = angle of max. elevation

The above formula is complicated and the fol-
lowing formula is usually sufficiently accurate

and takes into consideration as well the pinching
action between the guides and clips,

vi =^

W sin^gj+R

-)(lbs) where k = 1.1 to 1.2

i

«•••—

1

when ev is small as with symmetrically balanced
recuperator pulls, then Fvj = k[ Wr (sin0m+n coB/)B)+Rp]

where k = 1.1 to 1.2

If we include Rp with n Wr cos 0, we may in-
crease k, and we have the elementary formula as
before used, Pyi = 1.3 (Wrsin0B+0.3 cos0m) (Ibs)

704

Counter Recoil Buffer or Regulator Design

Counter recoil regulators may be divided in-
to two general types,

(1) Systems which are effective only
during the latter part of counter
recoil.

(2) Systems which fill themselves
during the recoil and are effective
throughout the counter recoil.

In type(l) we have a short spear buffer or
plunger entering the buffer chamber towards the
end of recoil. Type (1) buffers may be further sub-
divided into:-

(a) Plungers attached to a
continuous recoil rod, the re-
coil rod passing through a stuffing
box at either end of the piston.

(b ) Ordinary spear buffers with-
out a continuous recoil rod.

In the design of a counter recoil system, we
are primarily limited to a maximum allowable buffer
pressure, counter recoil stability in heavy artil-
lery being of no great importance since the stabil-
ity limit on a counter recoil is usually as great
as on recoil. Since, however, a considerable part
of the recoil energy becomes at the end of recoil
stored in the recuperator, we have this energy
absorbed in the counter recoil, by the counter re-
coil regulator in a short buffer displacement, with
a consequent large total buffer reaction. We are
limited in tbe counter recoil brake usually to a
smaller effective area than in the recoil brake;
consequently the buffer pressures become, due to
constructive limitations, very large. Hence it
is highly desirable to maintain as low a buffer
pressure as possible.

With any form of spear buffer of type (1),
to reduce the buffer pressure, the effective area

705

of the buffer plunger should be as large as pos-
sible and the length of buffer as long as possible.

In the design of a spear buffer of type (1)
we have the following limitations :-

(1) The diameter of the buffer,
should not exceed a value, that
due to the sudden withdrawal of the
buffer, the void displacement in
the recoil brake should not be
greater than the free recoil dis-
placement during the powder period
E.

(2) The length of the buffer should

not exceed a value that during the
counter recoil before the buffer

enters its chamber the buffer
chamber should be completely filled.
Let A * effective area of recoil piston (sq.ft)

A'= effective area of recoil piston on counter

recoil plunger side (sq.ft)
Lb=length of plunger or buffer (ft)
Ab * effective area of buffer (sq.ft)
du =diam. of buffer chamber

D - diam. of recoil brake cylinder
i_ = diam. of recoil brake rod
Now A 5 0.7854(Da-d») ; A'=0.7854 (D*-dg ) sq.ft.

Ab=0.7854(dg-d») (sq.ft) type (l)(a) buffer,

Ab=0.7854 dg (sq.ft) type (1) (b) buffer.
Now for condition or limitation (1), we have

(ft)

or A(Lb-E) g

A1 = A(l- -— )sq.ft.

Lb Lb

In terms of the diameters, we have

E
D«-dg

706

/£ E
)* T— + d*(l- — ) which gives us the

b limiting value of

db. It is interesting to note that when 6=0.
d]j*dr or in other words when the diameter of the
buffer a plunger is made equal to that of the rod,
no void is required in tbe recoil cylinder.

From the above expression, we note that in-
c re as ing the length of the buffer decreases the

diameter of the buffer and thereby increases the
buffer pressure.

On the other band the c 'recoil energy is ab-
sorbed over a greater distance with a longer
buffer, thus reducing the total buffer reaction,
and it is probable, that this cause more than
effects the slight increase of the buffer pressure
dueto tbe decrease of the buffer diameter. Further
the value of d|> is very often entirely limited by
constructive considerations alone; hence a long
buffer is highly desirable.

In a type (1) (a) buffer due to the relative-
ly large value of db required to give a sufficient
buffer area, the length of the buffer depends en-
tirely on the limitations (1). This type of
buffer will be considered in detail later.

For the limitation (2), with a continuous rod,
we have a void produced at the end of recoil on
tbe buffer side of the recoil piston. To compute
this void, we have, with an initial void in the
battery position AE, for tbe void on the buffer
side of the recoil piston at the end of recoil,
or the out of battery position.

Voidc=Arb-A(b-E=(cu.ft)

where Ar = area of the recoil cylinder (sq.ft)
Therefore, Voidc = (Ar-A)b + AE

= arb+AE(cu.ft)

Now in the c 'recoil, the spear buffer chamber
is evidently not filled until tbe void displace-

ment has been over run, and this displacement
a^> +AE .

v

becomes, Xa =

r

707

D*

AE
Since -— is small, for a close approximation, the

r buffer length should not exceed
Lb-0.8b(l- Jt )=0.8b(l-|f) ft.

The mean buffer pressure may now be computed,
knowing the potential energy of the recuperator.

The potential energy of the recuperator is
given by either of the following expressions:

pvivo
**o * fv ^B ~ D(ft.lbs) (k=1.3 approx.)

V

HO « - (ft.lbs)

Ar(k-D

where Vf=Vo-Avb

Av = effective area of the recuperator piston
(sq.ft)

Vo« the initial volume (cu.ft)

F?i = the initial recuperator reaction (Ibs)

%i
m » — • — = the ratio of compression

Fvi
Then, the mean buffer pressure, becomes

l»0-(Wrsin0+R0)b
Pb » *r - - - (Ibs/sq.ft)

where Ro=total packing and guide friction (Ibs)
b = length of recoil (ft)

d?
Lb = 0.8(1- ^-) (ft)

Ab = 0.7854 dg )sq.ft)

r-T' (ft'>

708

In type (1) (a) buffer, where we have a con-
tinuous rod and enlargement back of the piston for
the c 'recoil plunger, in order to have a sufficient
effective buffer area, the diameter of the plunger
must be necessarily large as compared with a spear
buffer. Therefore, to maintain a void displacement
in the recoil not exceeding the free recoil displace-
ment during the powder pressure period, we must
have a very short buffer. Hence if

A - effective area of recoil piston (sq.ft)

A'-effective area of recoil piston on c'recoil
plunger side (sq.ft)

Lb = length of plunger or buffer (ft)
effective area of the buffer

If further d« = diam.

of recoil brake rod
D = dia». of recoil brake cylinder
db = diam. of buffer chamber

we have db * CbD where Cb depends upon constructive

considerations

[(l-Cg)D«-dr]
and

Ab = 0.7S54(dg-d«)-0.785 (C£ D«-d»)

Now to reduce the buffer pressure it is de-
sirable to make Lb as long as possible and Ab as
small as possible. To do this we must make dr as
small as possible as compared with D. This re-
quires a large effective area for the recoil brake.

Hencs in type (1) buffer we may reduce the
buffer pressure by reducing the recoil brake pres-
sure. If HO = the total potential energy of the
recuperator we have

709

F - V

(m k _ x) (ft> lbs) (k=1.3 approx.)

= the ratio of compression

where Pyf

Av = effective area of the recuperator piston

(sq.ft)

VQ = initial volume of the recuperator (cu.ft)
then for the mean buffer pressure, ire have

W0-(Wrsin0+Rp)b
AbLb = 0.785

Now (D*-d»)(CgD«-d*)

- 2 L

(l-Cg)D*-d«

If we assume Cb = 0.7 roughly, we have AbLb=0.785(D2-d*
hence Ho-(Wrsin0+Rp )b

6 0.785(D*-d*)E
where

b = length of rscoil (ft)

E » free displacement in the recoil during

powder period (ft)

HQ = potential energy of the recuperator (ft.
Ibs)

RQ = total friction (Ibs)

since

we have pb = [B0+(WrsinJO+RQ)] -

PhE
where

Ph = total hydraulic brake pull (Ibs)

pb = assumed intensity of pressure in hydraulic

cylinder (Ibs/sq.in)

Therefore, to decrease the buffer pressure, with
a type (1) (a) c'recoil regulator:

(1) Lower the max. pressure in hy-
draulic braka cylinder during the

710

recoil.

(2) Decrease the length of recoil

(3) Decrease the potential energy
in the recuperator.

We see that the above expression is fixed by
the free recoil displacement E during the powder
period.

BY PASS PIPES In order to loner the buffer
USED WITH LARGE pressure on counter recoil, when
SPEAR BUFFERS. the c 'recoil regulation is by a
short spear buffer or plunger,
it is often necessary to in-
crease the diameter of the plunger materially over
that of the rod.

By the introduction of a by pass and valve
(closing on the counter recoil Heading from the
buffer side of the recoil cylinder to the outer
end of the void chamber of the buffer, the pressure
back of the recoil piston (on the buffer side) can
be effectively lowered without a full void by
being required in the recoil cylinder to take care
of the sudden withdrawal of the buffer plunger
during the first part of the recoil.

Let wa = required area of the by pass pipe

T * total powder period (sec)

tg=tirae of travel through void during the re-
coil (sec)

E = recoil displacement daring powder period
(ft)

A = effective area of recoil piston (sq.ft)

A'=effective area of recoil piston on plunger
side (sq.ft)

S * recoil displacement during void (ft)

Lfc - length of buffer (ft)

Pg = mean pressure in the rear of the recoil

piston (Ibs/sq.in)
p1 - max. pressure in the rear of the recoil

piston (Ibs/sq.in)

711

Now the total quantity of oil that Bust pass through

the by pass pipe, becomes, Q » A(L^-S)-A'L^ (cu.ft)
After the gun has recoiled the void displace-

ment, the void bade of the piston, i.e. the plunger
side of the recoil piston, becomes gradually filled

with the further recoil. The pressure in this
rear chamber however is zero until the chamber be-

comes completely filled. If Xs is the displace*
•ent in the recoil when this chamber is just filled',
obviously, A(XS-S)=A'XS hence

Xs = A^T7""

Let txs = the corresponding time in the recoil.
We have two cases:

(1) IThen Xs < E:

(2) Where Xs > E.

For case (1), txs and Xs are connected by the
equation,

pob (txs-to)*

Xs»Xfo*[Vfo- f- ""-'o)- 7-77-7-; KtM-t0)(ft>

r 6"r(Vf~7fo)

(approx)

from which by trial values we may estimate txs
For case (2) we aay compute txs from

2K(XS-E)

T * "here VX8 « V?

-xs K

~r

Vr » Vf approx. = max. recoil velocity (ft. sec)
K * total resistance to recoil
T = total time of powder period
To calculate the mean pressure in the chamber
back of the recoil piston, -

Dvl
pj = where D = density of the fluid-

53 Ibs/cu.ft.

vm = the mean velocity in the

Q pipe

and v. = — ; r ft/sec.

713

•here Q « A(Lg-3)-A'Lb

wa » area of pipe (sq.ft;

tb~txs = tine of travel through the recoil
displacement Lb~Xg

Now tb is the time for the recoil displacement
Lb, hence mr(Vr-Vb)

••here / 2K(Lb-E)
Vb » /?|

"r

To calculate the maxifflum pressure in the chamber
b.ck of tb. pl.t..,- D(,-A,)V.

™"* "' " — lbs/s<-ft-

where, when xg<E,

Vxs - Vf (approx)tbe maximum velocity

of free recoil (ft/sec)
where X, > E

— _— __««-.^— _

/ 2K(Xg-E)
vxs = ' Vt _

From the constrained velocity curve, we may cal-
culate p' during the displacement (Lg-Xs). Since
p >3po approx. we may assume p1 constant and use
the previous expression for pn. It is important
here to note that the recoil throttling must be
modified to maintain a constant pall on the brake.
Ph-pA-p'A1 (Ibs)

D Av oi"*

p-p'» (Ibs/sq.ft)

2gC'.«

Combining these two expressions, we may solve for
the required modified recoil throttling area wx
(sq.ft) in terms of the known values p' and Pb.

It is important that the recoil brake function
at least at the end of the powder. We place, there-
fore,

S » B, then Xs =

713

2K(Xa-E)
= ./ u* _
xs

mr
and tb = T+mr(Vr-Vb)

2K(LS-E)
Vb = A£ - • usually p1 should not

mr exceed a. few hundred

Ibs/sq.in. and the value

of wa and A1 should therefore be corresponding.
In such a case no material effect in the recoil
throttling is obtained and a modification of the
grooves is unnecessary.

DESIGN OF SIDE PRAMS The loading on the
GIRDERS. girders and the correspond-

ing stresses depends upon
the method proposed for
firing. These methods may
be classified as follows:

(1) Firing from semi fixed base

plate, with a large pintle bearing
and the girders extending to the
rear supported at their end by an
outer circular track. The horizontal
and a part of the vertical reaction

is transmitted to the pintle base

a. o i i -- ; ~J : ;;

plate, the horizontal reaction being

taken up by a vertical spade extend-
ing below into the ground from the
base plate and the vertical load
being balanced by the upward re-
action of the ground on the base plate.
No balancing moment is assumed to be

exerted by the ground on the base
plate. This assumption, makes it
possible to readily determine the
upward normal reaction of the outer
circular track. We have, therefor*,

714

with this method of loading the
horizontal and vertical reaction
at the pintle bearing and a vertical
reaction at the tail of the girder
balancing the trunnion reaction due
to firing. This loading should be
considered at both horizontal and
maximum elevation.

(2) Firing from the pintle base plate
assumed bolted down to a concrete
base. In this method no outer track

for supporting the tail of the girder
is necessary. We have therefore at
the pintle bearing a horizontal and
vertical reaction, together with a
bending couple balancing the firing
reactions at the trunnions. This
loading should be considered at both
horizontal and maximum elevation.

(3) Firing from a special layed track,
the nount recoiling in translation

on this track. By this method the
vertical load is somewhat distributed
by several shoes brought down in con-
tact with the track. The horizontal
component due to firing at the trunnions
is balanced by the total sliding friction
equal to the weight of the mount plus
the vertical firing component times
the coefficient of track friction
and the inertia resistance of the
mass below the trunnions to ac-
celeration. Though the horizontal
reaction on the trunnions is theoretically
slightly reduced due to the acceleration
of the cradle in which the gun recoils,
we may practically consider that the
total firing load is brought on to the

715

trunnions, since the acceleration
of the total mount backwards is
relatively small and the mass of

the cradle quite negligible as com-
pared with the large mass of the
main girders, trucks, etc. below
the trunnions. The vertical com-
ponent due to firing at the
trunnions is balanced by the upward
reactions on the various shoes.
Finally the couple produced by the
horizontal reaction at the trunnion
and the resultant of the inertia
resistance and the shoe frictions,
is balanced by a couple produced
by the vertical reaction at the
trunnions and the resultant normal
or vertical reactions of the track
or guides on the various shoes.

This requires a uniforaiity increas-
ing upward reaction on the various

shoes towards the rear. The load-
ings should be considered at

horizontal and maximum elevation.
(4) Firing directly from trucks

riding or recoiling back on the rails,
This loading is similar in character-
istics to (3) except now the sup-
porting reactions are concentrated
at the truck pintles. Again the
loadings should be considered at
horizontal and maximum elevations.
When a girder is designed to meet all four
requirements in the methods of firing, we have for
the two elevations, eight types of loading to be
considered as applied to the girder. Knowing then
the loads brought on to the girde, we have, the
following points to consider in the layout of the
girder as regards its strength.

716

(1) The proper flange area to carry
the requisite bending at a section
of given depth.

(2) The proper depth of girder for

all other sections.

(3) The proper cross section of the
webs for carrying the total shear.

(4) The proper pitching of the
rivets for carrying the longitudinal
shear .

(5) A careful study of web reinforce-
ments or stiffeners.

(6) The distribution and design of

cross .beams or transoms connecting
the two girders.

(7) The detailing and design of the
pintle bearing.

(8) The reinforcement in the web re-
quired for the elevating pinion
bearing.

Reactions between tipping parts and girder

trunnion reactions:

2H=K cos 0+E cos 6^ (Ibs) (1)
2V-K sin 0-E sin 6^ +Wt (lbs)(2)
and for the elevating gear reaction.

Ks+Pbe

E = ; In battery (Ibs) (3)

J

Ks+»rb cos 0

E * Out of battery (Ibs) (4)

j
where

H and V * the horizontal and vertical com-
ponents of the trunnion reaction
(Ibs)

K = total resistance to recoil (Ibs)
E = elevating gear reaction (Ibs)
j » radius from trunnion axis to line of

action of elevating gear reaction - with
rack and pinion = radius of each (in)

717

6£ = angle between j and the vertical.
S » perpendicular distance from line through
center of gravity of recoiling parts
and parallel to bore to center of trunnions
(in.)

e = perpendicular distance from axis of bore
to center of gravity of recoiling parts.
With a balancing gear introduced between the
tipping parts and girders, we must modify the
trunnion reaction to include this reaction. The
elevating gear reaction is not changed, since
the moment of the tipping parts about the trunnions
is always balanced by the balancing gear in the
battery position of the recoiling parts.

Since it is usually customary to locate toe
trunnions along a line through the center of gravity
of the recoiling parts parallel to the bore, S =0,
and therefore, p e

E = "-; — in battery

/X X>0 \
Wrb cos 9

g m . out of battery
j

Now since e is usually made very small,

Pbe Wrb cos 0

and may be neglected as compared with

J j H and V. Hence, we will assume

the elevating gear reaction to

be negligible, and we have the total firing load
brought onto the girders at the trunnion. Then

2H= K cos 0 : approx. reaction between
2N=K sin0+Wt : tipping parts and girder.

Reactions between base plate and girder.

Considering the reactions on the base plate,
if it is considered that the ground can offer no
bending resistance as in assumption or method (1)

718

OF

9.1-f

. e>

Fig.

719

1

A/

^-;

//-* — ta

\*-*7^Vv~

...... 1 \AS.

1

S^ VVy

X

^ VA>

rfr, »,

3 ^^

i

-c
i

-c

J

J7^^ .""Lfl-

T

i..l

Fig. 3

720

of loading we have the reaction between the base
plate and girder as equivalent to:-

(1) A vertical reaction through
the center line of the pintle

bearing * V (Ibs)

(2) A horizontal reaction at the
pintle Hp (Ibs)

(3) A couple Hp(h-hp)(in.lbs;
where h = height from ground to trunnions (in)

hp = height from pintle bearing to trunnion

(in)

In method (2) of loading we have the reaction
between the base plate and girder equivalent to:-

(1) A vertical reaction through the
center line of the pintle bearing?

VP

(2) A horizontal reaction at the

pintle H
P

(3) A couple resisting the over-
turning moment * H h

P

Constructively, only the horizontal reaction
is taken up at the pintle bearing, the vertical
or normal reactions being taken up at the travers-
ing rollers. Thus, the roller reactions are
equivalent to a couple H hp and a resultant vertical
or normal reaction V

To calculate the individual traversing roller
reactions we proceed as follows:

Consider the rollers equally spaced
around the periphery of the roller path. Then,
taking loooents about the front outer or end
roller in the direction of the axis of the bore,
we have, for the various roller reactions, see
fig. (2).

Assuming "n" chords passing through a pair
of rollers and perpendicular to the axis of the
bore projected in a horizontal plane, then,

721

pt=k(xt+y ) pa»k(xa+y ) ---- pn=k(xn+y)
Taking moments about the front roller
k[2xt(xt+y )+2x2(x2+y ) --- 2xn-i+y=+xn(*n+y>

= H hp+Vpr
Simp lif ying, we have

ky(2xt+2x2 -- 2xn_1+xn)+k(2x*+2x2 --- 2x*_1+x*)

- H hpvVpr

and for the summation of the vertical reactions,
ky+Wk(xt+y)+2k(x?+y) -- 2k(xn_a+k(xn+y )=Vp

2k ny+k(2xt+2x2+ --- 2xn_1+xn)=Vp

To solve, we note that, A(ky)+B(k)=H hp+Vpr

C(ky)+D(k)=V
where A =(2xi+2xs --- 2xn_x+xn)

B =

C = 2n
D =(2x

Knowing x^x --- xn we may readily obtain Po»Pt --- pn
To compute x^x -- xn for the rollers, we bave

for the angle to the various chords,

2n 360

6 * — radians or - degrees
n

_ 2n 360 .

9 -2 — rad. or 2 - degrees
a n n

n 2* n 360
9n = 2 ~ rad' °r 2 ~

tbereforex»r(l-cos 6^) (in)

xa=r(l- cos 92) (in)

xn= r(l- cos 9n) (in)

where r = radius to the center line of the roller
path.

722

Proa the previous equations we nay now compute P0,
Pt - Pn(lbs), the individual roller reactions.

The previous formulae, assume contact betiieen
each roller and the roller track under maximum
firing conditions. If the roller path has a small
diameter, we nay have the condition, when, only
the rear roller is brought into contact, the over-
turning moment on the girder being balanced by
a couple exerted by the base plate an upward re-
action at the rear roller contact and a downward
reaction at the front circular clip contact. If
the circular clip has a radius approx. equal to
that of the roller path, then we have for the
sax. roller reaction Hphp+Vpr=2pmaxr

H VV
.av* - where r = radius of the

roller path (in)
P>ax = max. roller reaction
(Ibs)

Vp=aax. upward reaction at
pintle (Ibs)

External forces exerted on the girder during

firing:

The external force or the girders are
shown in plates A and B for the four methods of
loading.

In method (1) of loading, we have the re-
actions of the tipping parts H and V, the reaction
of the base plate H and V together with the
couple Hp(h-h_) and the reaction of the outer
track on the tail of the girder N. Further we
must include the total weight of the girder which
though actually distributed we will assume con-
centrated at its center of gravity at horizontal
Ig from the axis of the trunnions.

723

Taking moments about the pintle bearing,

H h +H(h-h_)-Nl =0 hence

v * Hh

N = ^ (Ibs)

*n

where H = K cos £5 and h = the height of the
trunnions from the ground (in).

Knowing N we may compute for the strength
of the tail of the girder, for method (1) of
loading.

In nethod (2) of loading since we are detail-
ing the strength of the girder in the region of
the trunnion and pintle reactions, we must take
the actual components of the reaction into con-
sideration. These consist of the trunnion and
elevating arc reactions of the tipping parts, that
is the reactions H,V, and E and the reaction of
the base plate consisting of the various roller
reactions and the horizontal reaction of the pintle
as shown in "Reaction of Base Plate on Girder" diagram,

In method (3) of loading, where the mount slides
back on a special constructed track, we have for
the reactions on the girder.

(1) The H and V components of the
trunnion reaction of the tipping
parts.

(2) The inertia resistance of the
girder, resisting tne acceleration
of the girder acting at the
center of gravity of the girder =

dt«

(3) The weight of the girder acting

at its center of gravity Wg

(4) The normal reactions of the track
shoes Na and Nb

(5) The frictional or tangential com-
ponents of the track shows n(Na+Nb)

724

In calculating the stresses on the various
portions of the girder we must of coarse consider

both the weight and inertia as distributed forces,
but for dealing with the overall reactions, we

may assume their resultant effect as concentrated
force passing through the center of gravity of
the girder.

When the trucks are entirely disengaged in
this method of firing, we have,

d2x
H-nUa+M-"^ dT1 = ° 3nd Na+Nb=v+V*g when tlie

trucks are

not disengaged but hang from the girder, we must

consider both their weight and inertia reaction,
hence if Wtjj = weight of trucks (Ibs)

Mt- = mass of truck
we have,

H-n(Ka-KTb)-(o)g+mtk) - and Na+N
dt*

To compute Na take moments about Nb (see fig. (4).

d*x
*.(Vlb) + H h-Vlb-Vlb-Wg(lb-lb)- mg — (h-hg)-0

hence H*X

)+« ~(h~ " H *

N = - — -- - (Ibs)
a la+lb

and for Nb talcing moments about Na, we have

- .

(h-hg)

hence

„ , - - (Ibs)

In method (4) of loading, we have the mount
recoiling back directly on the rails, and the
trucks react on the girder with reactions Ha,
N and Hjj, Nb, at the truck pintles a and b . The

725

tipping parts react on the girder with components
H and V at tne trunnions. In addition we have the

inertia resistance

d8x
nig - — - resisting the ac-

celeration and the

weight of the girder both acting through the center
of gravity of the girder.

For a horizontal motion back along the rails,

tie have

d x
H -(H.+Hv)- m- - — - = 0 and normal to the

** U t %*.* I

rails, V+Wg-(VNb) = 0

To calculate Ha and H^ the horizontal components
of the truck reaction we must consider the trucks
separately. In firing directly from the rails
the trucks are usually braked.

If W^jj and M^ = weight and mass of either truck
Ww and f»w = weight and mass of a pair of wheels

I=«wka = moment of inertia of a pair of wheels

about the center line of the axle.
d = diameter of a car wheel

k = radius of gyration of a pair of car

wheels (=0.7 d approx.)
NW = normal reaction at base of car wheel

N«. = normal reaction of brake shoe on wheel

»*

per pair of wheels

fw = coefficient of rail friction
* fs = coefficient of brake shoe friction

Rw=tangential force exerted by rail on base

of car wlieel
Now for the motion along the rails, we have,

dax
Ha - S Rw = mtk j^

Considering the rotation about the csnter of gravity
of a single wheel we have,

2mwk2 d2x 4mwk2 d2x

2Nsfs wbere n = no. of

pair of wheels per

726

(3) Of LOAD/NG

1

te.« v<— j

I M A//*

MfT/iOD (4)0f£ O4D/MG

1 I/
^I^K

• ^^-£=

A/-

s-\ ^T

•f X-^HH —

§

-9++

Fig. 4

727

SfCT/OA/ A-B

o o o

O O O

T

I

e-j ^* I:

Fig. 5

728

4rawk» d*x
truck and likewise Hb=(mtk+n — — )j^+SNsfs

The tera 2Ngfs is difficult to calculate since it
depends upon bow hard the brakes are set. If the
brakes are set to skid the wheels, no rotation
occurs, and we have (jax

ZN f

Assuming fB=0.2 and 2Nw=Hg+V+Wtk we have,

dax
Ha+Hb=2ntk — + 0.2 ("g+V+Wtk) and therefore

d*x H-0.2(Wt+Y+»tk)

- = - froffl which we may easily

dt» 2mtk+mg calculate the horizontal

inertia loading for any position of the girder.

The reactions at the truck pintles, become

res

spectiveiy,
b* g^ b 1g' +

dax
mg (h-hg)-Hh

/i u _ \

d2x

'Btk * ~+Q-2 Na
dt2

Ubs;
(Ibs)

*b

Hh+Vlj+Wgdg+1

5 * d t 2 _ .» flVio^

\liOS )

+0.2 Mb (Ibs)

Comparison of Truck Pintle Reactions.
In method (3) and (4) of loading we find

Nb-Na = -___—— Now in general the

horizontal resistance
is small as compared with the inertia resistance

nu — r Hence we may approximately assume H=mg— 7

8 dt 2H(h-hg) Kdt

therefore Nb-Na = Further, we are not

la+^b greatly in error in

729

assuming hg= - then we have, Nb-Na = " i —

1a*1b

That is the difference of load thrown on the rear
and front truck respectively equals the horizontal
trunnion reaction times the height from the
trunnions to the horizontal center line through
the truck pintles, and divided by the distance
between the trucks.

Obviously as the gun elevates H decreases,
while V increases; therefore at max. elevation
the loadings on the trucks are more nearly
equalized.

With railway carriages, since at maximum
elevation

h

H is relatively small compared

with Na or Nb, for all

practical purposes we may consider that the re-
quired strength of the girders must be equally
strong on either side of the trunnions.

CHAPTER XI.
GUN LIFT CARRIAGE.

Single recoil systems where the recoiling
mass does not translate in recoil parallel to the
axis of the bore, appear in various types of mounts.
Illustrations of such types nay be found in our
model 1897 Barbette mount, where the gun and top
carriage fora a single recoiling mass, recoiling
up an inclined plane. Railway carriages especially
in France bare been used, where the recoiling mass,
(gun and top carriage ) recoil on a gravity plane
mounted on the car. The object of the inclined
plane is to return the piece by gravity into
battery. Carriages with no recoil except the slid-
ing back of the gun and top carriage as a single
mass on rails have also been extensively used, the
resistance to recoil being merely the friction
offered by the rails or slides.

CHARACTERISTICS OP Due to the fact that the
INCLINED PLANE recoil is not along the
CARRIAGES. axis of the bore, during the

powder period, a component
of the total powder force
normal to the inclined plane

or slides is introduced. This component therefore
introduces large stresses in the carriage, the
component increasing with the elevation. The ex-
cessive stresses thus introduced at high elevation,
prohibits the use of this type of mount for firing
at high elevations especially for large calibers.
The type of mount is useful for where the elevation
is not great. With large size howitzers this type
of mount would necessarily produce a very heavy
mount for strength and, therefore, from the point
of view of mobility alone could be regarded as
none else than poor design.

732

Since the gun recoil is not along the axis
of the bore a reaction on the projectile normal
to the bore is introduced. This reaction reaches
a maximum closely at the maximum elevation. It
possibly introduces unequal wear on the rifling
in the gun tube itself. This reaction further
introduces a slight spring during the powder
period on the elevating arc and pinion.

APPROXIMATE THEORY OF Even, for a very close
RECOIL, NEGLECTING approximation the reaction
NORMAL REACTION OF of the projectile normal
PROJECTILE ON BORE. to the bore during the

powder period has a very

snail effect on the recoil, though it is of
importance in estimating the maximum elevating
arc reaction during the powder period. If,
then we let

Pfc = total powder reaction on base of projectile,

in Ibs.
B = hydraulic braking of recoil cylinders

parallel to inclined plane in Ibs.
R = total friction of the recoil in Ibs.
wr and mr = weight and mass of recoiling

parts (in Ibs)
0 = the angle of elevation of the axle of the

bore

6 * the inclination of the inclined plane.
E = displacement of free recoil during

powder period (in ft.)

T * total time of powder period (in sec.)
Vf= velocity of free recoil (in ft/sec)
K = the total resistance to recoil, in Ibs.
b = length of recoil, in ft.

Then considering the recoiling parts during the
powder period, we have,

u V

Pbcos(0+6)-(B+R+Wrsin9)= «r r— and since

dt

K»B+R+WrsinO

733

734

Pb cos(2J+9)dt
then /

"

r

but Phcos(0+9)dt

/— - = Vf cos (0+9)
n>r

therefore at the end of the powder period, we find

KT

Vr=Vfcos(0+9)- — (1)
mr

KT8
and Xr'J) cos (0+9)- - — (2)

2n j,

During the remainder of the recoil, we have

j mrV* = K(b-Xr) (3)

Substituting (1) and (2) in (3) and simplify-
ing we have

£ mrVfcos2(0+9)

K = - (4)
b-(E-VfT)cos(0+9)

Obviously Vfcos(0+6) and E cos (0+9) are the
component free velocity and displacement parallel
to the inclined plane.

EXTERNAL REACTIONS ON THE If we consider the sys-
RECOILING PARTS AND TOP tern, of the gun wg and
CARRIAGE ROLLER RE- recoiling top carriage
ACTIONS. wc, we have by D1 Alemberts '

principle, considering

inertia as an equilibriating force, the following
external reactions:-

(1) The powder reaction along the
axis of the bore — Pb

(2) The inertia force of the re-
coiling mass, opposite to the
motion during the acceleration,
and in the direction of the
motion during the retardation

and parallel to the inclined plane —

d*x

•r

735

(3) Weight of the total recoiling
parts Wr

(4) The normal reaction of the
rollers E N

(5) The braking pull exerted along
the axis of the hydraulic brake
cylinder B

(6) The total friction along the
roller track R

These forces are shown in fig.(l)

Pesolving (1), into a couple and a single
parallel force through the center of gravity of
the recoiling parts and combining with (2) we
have,(l) and (2) equivalent to,

A powder pressure couple Pbd

where d = the perpendicular distance between the

center of gravity of the recoiling parts

and the axis of the bore .
A component parallel to the inclined plane

through the center of gravity of the recoiling

parts dy

Pbcos(0+9)mr — =B+R+Y»rsinp=K

and a component normal to the inclined plane through

the center of gravity of the recoiling parts

Pbsii)(0+9) Thus (1) and (2) reduce to

A couple Pbd and the parallel and normal com-
ponents through the center of gravity of the re-
coiling parts, K and Pbsin(0+9)

To reduce the couple Pbd and the consequent
stresses, the center of gravity of the recoiling
parts should be located at the axis of the bore,
or slightly below to ensure a positive jump.
Since the center of gravity of the gun is at the
axis of the bore, the top carriage center of
gravity should also be located at the axis of
the bore. This is impractical, but if the top
carriage is made light as compared with the gun,
its effect in lowering the center of gravity of

736

B

I1 'I' T 'i

737

the total recoiling parts is small.

To compute the roller reactions on the' in-
clined plane, we proceed as follows:

Taking moments about tlie front roller
reaction "0", we have Khr+Pbd+Pblrsin(0+9)«fWr (lr

cos9-hrsin8)-Be=Nilj+N 1

---- Nnln (5)

where hr and lr are the coordinates normal and
along the inclined plane of the center of gravity
of the recoiling parts with respect to the front
roller "0"

e = the moment arm of B with respect to "0"
Nnln = the moment of the n th roller reaction
about "0"

When the top carriage is light as compared
with the gun, the center of gravity may be assumed

approximately at the trunnions and therefore P^d^O
Hence (5) reduces to, Kht+P^ltsin (flf+6 )+Wr (lt

cos6-htsin 6 )-Be* N^+N^ ---- Nnln (6)

where htand lt are the coordinates of the trunnion
with respect to the front roller "0". Further,
we have, Pbsin (0+0) +Wrcos 9 =No+Ni+N2 ---- Nn (7)
If we assume the roller base is rigid, we "have
B«k(l+c) N

Therefore if, SMQ= Htlt= N2lg ----- Nnln

ZN = VV*. -------- Nn

we will have, MQ=T<(1*+1* + 1» ------ 1«) kc(li + lf + l3 -- ln)

(8)
ZN*k(l1 + la ------ ln) + (n+l)kc

From which we determine "k and c

EXTERNAL REACTIONS ON THE If we consider the
MOUNT AND TRAVERSING ROLLER system consisting of
REACTIONS. the gun, and top car-

riage, that is the
recoiling parts, to-

gether with the "bottom carriage which rests on a
circular base plate supported by traversing rollers,

738

we nay eliminate the Mutual reaction between the
recoiling parts and bottom carriage since it has
no effect on the equilibrium of the system. Further
by the use of D'Alembert's principle we may again
regard the inertia resistance of the recoiling
parts as an equilibriating force.
We have therefore as before,

(1) The powder pressure couple Pbd

(2) The total resistance to recoil
through the center of gravity of
the recoiling parts and in the di-
rection of the recoil K

(3) The weight of the system Ws

(4) The pintle reaction balancing

the horizontal component of (2)

(5) The traversing roller reactions.
Let Ws= weight of system

13 - moment arm of Ws in battery about rear

traversing roller
lg = moment arm of HS at recoil X or b from

battery

Wbc = weight of bottom carriage
lbc = moment arm of tf^c ?bout rear traversing

roller

Wr = weight of recoiling parts
l^. = moment arm of Wr in battery about rear

traversing roller
b = length of recoil

The moment of the weight of the system
changes during the recoil. If we take moments
about the rear traversing roller, we have for the
weight during the recoil WpdJ-Xcos 6 )*Wbclbc=«sls
hence Wslg= Wsls - HrX cos 6 and when X = the
length of recoil b, we have Wslg = Wsls-Wrb cos0

Further if, h^ and 1^1 are the vertical and horizontal
battery coordinates of the center of gravity of the

recoiling parts with origin at the rear traversing
roller then the out of battery coordinates become

and (l£-b cos 9) respectively. We have

739

for the nonents about 0, in battery W_l_-PKd-Khi

_-_._--- 8 S D

cos 6 - K lr sin 6 + Pbsin(0+e)(i£cos 6- hr
sin 6)+2Nilj+2Nslt Nnln (11)

and in the out of battery posit.ion

Wgls-Wrb cos e-K(br+b sin 6)COs 6-K(kr- b cos 6)sin 6

* 2 Ni1^2N2la Nnln (12)

If we assume the center of gravity of the recoil-
ing parts at the trunnions, then Pbd disappears,
and h£ = h{ and 1^ » 1{ As before NQ«kc, N =k(lt+c)
Nn=»k(ln+c) hence

9 II 1 j. O W 1 Ml -l*fO12j.Ol2 1»\ / 1 *5 \

2Ni1t +2Na1. W« * k(2 1i*2 ll ' ~1V (13^

We also note that PbSin(0+6)cose-Ksin 6+l»8= EN (14)

where EN=k(2 lt+2 12 (21n_1+ln)+2kcn

From equation (13) and (14) we may solve for k,
and c and thus eonpute the roller reaction NQ,^

INTERNAL REACTIONS With gun lift mounts the
TRUNNION REACTIONS, trunnions are a part of the

gun itself and are located
at the center of gravity of
the gun. Neglecting the

normal reaction of the projectile, and taking
moments about the center of gravity of the gun,
that is about the trunnions, we have, E j = 0,
(j-eonent arm of E about the trunnions), there-
fore the elevating arc reaction E = 0. If Xt
and Yt are the components of the trunnion re-
action, parallel and normal to the inclined
plane, respectively,

w- = the weight of the gun alone. We have,
considering the gun alone, fig.(

dV
)-WgSin 9 - mg — -

I

(15)
2Yt=Pbsin(0+6 )+Wgcos 6

dV

but Pb cos (0+8) - K = mr -

dt

740

dV "a ag

hence ng — = — Pb cos (0 + 9)- K —

1 dt mr mr

Substituting in (15), we have

°0 "ff

2X . p. cos (0+9K1 ^)+K -*• - Wesin 6 1

flj ID

\ (16)

9

which gives us the components of the trunnion
reaction. The resultant trunnion reaction,
"becomes,

St - / X£+Y$ (17)

The elevating arc reaction is zero, except
during the first part of the powder pressure

period.

To compute this "whipping action" during
the powder pressure period, we must plot the
moment of the normal reaction of the projectile
about the trunnions as the projectile moves along
the "bore.

The normal reaction of the projectile,
equals,

N =m — sin (01-6) (18)

dt

The weight component normal to the bore "being

neglected since we will assume a fairly large
breech preponderence, but

dv P^cos (0+6)-K Pbcos(0+9)
dt mr mr

If U = the travel up the bore

Ut * the distance from the center of the
projectile in its initial position
to the center of the trunnions.

Then, the elevating arc reaction becomes,

N (U-Ut)
E » -*

j

741

«Pb(U-Ut)sin2(0+9)
= - - - (19)
2<"rj

From a plot, the maximum moment was found to oc-
cur, when the shot reaches the muzzle, and we

then have for the maximum elevating arc re-

action, mPob(U0-Ut)sin2(0+e)

E = ° - (20)

breech when shot
leaves muzzle)

n iic27 P|°ax ,» /,, 27 pmax.

C = U( --- 1)+ / (1- -~ - )«-i * (twice

16 pe 16 pe

abscissa
cf max.

pressure)

Pe = (pjj =total max. powder force on

breech)
VQ = muzzle velocity; Pe= mean powder reaction

on breech
Ifr - travel up bore in feet

REACTIONS ON TOP Neglecting the elevating arc
CARRIAGE. reaction during the powder period,
the reactions on the top carriage

reduce to the following:—

(1) The trunnion reactions divided

into Xt an:* ^t an<* equal and
opposite to the component re-
actions exerted on the gun.

(2) The weight of the top carriage
acting through its center of
gravity --- Wc

(3) The braking pull reaction --- B
(4) The roller reactions of the in-

clined plane.

Assumng the center of gravity of the top carriage
at the trunnions for convenience, we have

dV
2Xt-Wcsin 6 - B = mc —

742

2Yt+Wccos 6 = £ N (22)

and taking moments about the front roller reaction,
we have 2EMQ = 2Xtht-2Ytlt+Wccos 6 lt-Wcsin 6.ht

dV
- Be - mc — ht (23)

dv [Pbcos(0+9)-K]

where " s ~

" at ror

ht and lt are the coordinates of the trunnion
with respect to the front roller
"0", and normal and parallel to
the inclined plane.

9 = the perpendicular distance from the front

roller to the line of action of B
If, as is usually the case, the center of gravity
of the top carriage is not located at the trunnions,
we have equation (21) and (22) the same, but equation

(23) modified to:- Pbcos(0+8)K
M0.2Xtht+2Ttlt+Wccoa 6 lc-Wcsin 9 hc -[— ]

»chc-Be (24)

where lc and hc are the coordinates of the center

of gravity of the top carriage parallel and normal
to the inclined plane and with origin at the front

roller. As before, the moment of the roller re-
actions 2Mo=Nt1i+N81a'l'Ns13 Nn1n

therefore 2Mo=k(l»+l|+l* l*) + kc (li + la+la ln)

SN = klt+kla kln+(n+l)kc

and Nn=kc, N »k(l + c), N =k(l +c) N-^kdn+c)

O ' I 1 ' 2 fc

that is solving for k and c we determine Nt S^
Nn knowing the total normal.

Substituting in Eq.(24),
"g mg
mr mr

2Yt aPbsin(0+e)*wgcos e and noting that, niglt+mclc»

mrlr

743

mght+IDchc = mrnr we have,Pb[ (ht-hr)cos(0+e)

cos 6 - Be = 2 MQ (25)

Now (ht-hr)cos(0+9)+(lt-lr)sin(0+9) is evidently
equal to the perpendicular distance between the
center of gravity of the total recoiling parts
and the axis of the bore. Hence (25) reduces to
Pbd+Pbsin(0+9)lr+K hr-Wrhr sin 8 + Wrlr cos 6 - Be

= Z M0 (26)

where d =(ht-hr )cos (0+6 )+(lt~lr )sin(0+9)
This is evidently the same as equation (5) obtained
in the consideration of external force on the re-
coiling parts.

REACTIONS ON BOTTOM CARRIAGE. The reactions

on the bottom
carriage consist
of the following:-

(1) The braking pull exerted along
the axis of the hydraulic recoil
cylinder.

(2) The roller reactions normal to
the inclined plane.

(3) The horizontal reaction exerted
by the pintle bearing.

(4) The supporting reactions exerted
by the traversing rollers in a
vertical direction.

Evidently (1) and (2) is the reaction of the top
carriage on the bottom carriage, which is divided
into the components (1) and (2).

Thus in battery, the moments of (1) and (2)
about "0" the point of contact of the front roller
reaction of the inclined plane reduce to ZMo+Be
but ZM0+Be=Pbd+Pbsin(0+9)lr+Khr- Wrhr sin 8+»rlrcos 9

where ir and hr are the coordinates of the center
of gravity along and normal to the plane of the re-

744

coiling parts with respect to the front roller.

Therefore during the powder pressure period
the reaction of the top carriage on the bottom
carriage is equivalent to,

(1) A powder pressure couple "Pbd"

(2) A component of the powder force
normal to the inclined plane and
through the center of gravity of
the recoiling parts nPb sin (0+0)"

(3) The total resistance to recoil
parallel to the inclined plane,
and through the center of gravity
of the recoiling parts "K"

(4) The total weight of the recoil-
ing parts through the center of

gravity of the recoiling parts

ii ui "
wr

During the pure recoil or subsequent retardation,
we have, 2M0+Be=Khr-Wrhrsin 9+Wrlr cos 6
and therefore the reaction of the top carriage on
the bottom carriage, is equivalent to

(1) The total resistance to recoil
parallel to the inclined plane
and through the center of gravity
of the recoiling parts K.

(2) The total weight of the recoil-
ing parts.

To compute the horizontal pintle reaction, we
have H » K cos 0 -Pb sin(0+9)sin 0 the total
normal reaction on the traversing rollers, be-
come ZN*Ptsin(0+9)cos 9 - K sin 9 +1"r'f*bc
where W^c = weight of bottom carriage
If further 1£ * moment arm of »r in battery about

rear traversing roller
x « recoil displacement from battery
lbc= "oment arm of Wbc about rear traversing

roller

Ws = weight of entire system above traversing
rollers

745

lg= moment arm of Wg about rear traversing

roller

Then, for the moment of the weights about the rear
traversing roller, we have,

Wr(lp- x cos 0)+*bc1bc = lf's1s ~ wr x cos ^
Therefore, for the moments about the rear travers-
ing roller, we have
ZM0=Wsls-Wrx cos 0-Pbd+Pbsin(0+6)[ (l£-x cos6)cose

-(h£+ x sin8)sin Q] -K( (h£+ x sin8 )cos9-(l£-x cos6)sin8]

When Pb is a maximum x is negligible; therefore for
the maximum roller reaction, we have

ZM0= Wsls-Pbd+Pbsin(0+9)[l^cos8-h;sine]-K[h^cos6-
EXACT THEORY OF RECOIL Doe to the normal reaction
CONSIDERING NORMAL of the powder charge and
REACTION TO BORE OF projectile during the travel
PROJECTILE. up the bore, the recoil is

more or less effected, de-
pending of course on the weight of the shell and
powder charge as compared with the weight of the
recoiling parts. Let

Pb = powder reaction on breech of gun
P_= powder reaction on base of projectile
Pe - mean powder reaction in bore of bin
N ^normal reaction of projectile to axis of

bore
N"t= normal reaction of powder charge to axis

of bore

N = N + N = the total normal reaction of
powder charge and projectile
to axis of bore.
B + R = total braking resisting recoil

parallel to inclined plane,
w and B = weight and mass of projectile
wr and mr = weight of mass of recoiling mass
w and m = weight and mass of powder charge
0 = the angle of elevation of the axis of the
bore

746

9 * the angle of inclination of the inclined

plane
x1 and y* = coordinates along and normal to

tbe axis of the bore.
a = travel of the projectile along tbe bore

or relative displacement along tbe axis

of tbe bore
x = tbe projection or component of the absolute

displacement of the projectile parallel

to the inclined plane
Considering the motion of the projectile, we have

d*x '
Pp - n — + ng sin 0

N = • - sin (0+6)+«g cos 16 (2)

dt»

where

d«x' d«u d«x ,., -. ,_.

T - T - ? ««• <«*•) «)

for the motion of the powder charge,

" PP = [ ~ 2 cos(0+e>1+5« sine)

N
*

ird*x' d*x
= ytr—; -- r-T CO8 (0*8)k»g sin «J (4)

* dt* dt

5 ^ sin(0+e)*Ig cos A (5)

4 1

where

. 2

(6)

Is the resultant acceleration of the center of

gravity of the ponder charge, and for tbe action

of tbe recoiling parts,

Pbcos(0+e)-N sin(0+e)-«rg sin 6 -(B+R)-«P ^

Nbere N « Ht+N,

Combining the above equations, we have

747

I.d«x' I d*x

(«)•*•-; ... cos(0+o;- - T—rcos2 (0+6 ) + (m+m)g sin0cos
e, at* 2 dt*

d*x

(0+9)-(m+I) sin* (0+9)-(m + [B)g cos0sin (0+6 )-ni_g sin9

dt«

d«x
-(B+R)=»r —

Expanding and simplifying, we obtain
• d'x1 5 dgx _ I d'x .

dax
-(m+I+iBr)g sin 9 - (B+R) * mr — - -

U t-

that is

n . ,d*x * d*x . m. dflx

=B+R*(B+I+«r)g sin 9 (8)

It is to be noted that

d*x' d*x

[ cos (0+9)- -—rsin* (0+9)] is the projection or
dt * dt

component of the re-

suotant acceleration of the projectile parallel
to the inclined plane, and B+R+(m+mr+in)g sin 9
is the total external force parallel to the plane.
Neglecting gravity and with free recoil(B+R=0)f
that is no extraneous force acts, hence we have

Ma

In terms of the relative acceleration *— *

d t

d»x' d»u d»x .
since

i.d"u . d«XT . m.d*x

[ (.*- Jcos (0+8 ) +— ] » (•+)

hence

(*n+")T-T cos(0+9)=(m+B+»r)7f7 (10)

2 dt2 dt1

and by integration

<•*? )J7T cos (0+9)=(m+iii+inr)^ (11)

cat a t

748

(a+-)u cos (0+9)=(a)+m+mr)x (12)

Hence (10, (11), and (12) gives us the free ac-
celeration, velocity and displacement up the inclined
plane with respect to the corresponding function
up the bore of the gun.

Again considering equation (8) and substituting

d«x' d«u d«x

— — = - - — cos 10+9)
dt« dt« dt«

we have

m d*u d*x

(m+«)T~7 cos(0+9)-(B+R+lm+fii+mr]g sinQ )-(m+m-»-in..) — r = 0
2 at* * d t

hence

dtx dtu B+R(m+S+ni )g sin9

(13)
dt* n+I+mp

Integrating,
+

du B+R+(m+m+mr )g sin 6
— =( - )-- coa(0+9)-[ - = - ]t (14)
dt

and

m

B+R+(m+i+mr)gsin 9

x = (— = - )u cos(0f9)- -[ - = - ] t* (15)
B-HB + nij. m+m+mr

which are the general equations of constrained re-
coil during the travel of the projectile up the
bore.

Neglecting m and ro as small compared with mr
and if we let m

then

dx B+R+mPg sin 9

— » Vf co8(0+ot)-( - - - )t

dt a

B+R+Bpgsin6

x = E cos (0+«) - i ( - 1 - ) t* (15')

"r

which are sufficiently approximate for ordinary

749

calculations .

HUMERICAt COMPUTATION.

10" Gun Barbette

K - braking force

cos (0+6)*

(0+8) * 190

2 b+T Vfcos(0+6)-E cos(0+6)

89.820(29.76x0.9455)'

2 (—+0.0446x29. 76x0. 9455-0. 9455)32. 2

L &

247000 Ibs.

89 . 820 (29 . 76x0.9976)'
5Q — • 274000 Ibs,

32.2x2(— +.0446x29.76x0.9976-0.9976)

1 o

Wr » 89820 Ibs.
Vf * 29.76 ft/sec,
b * 50 in.

T » 0.0446 sec.
E » 1 ft.

Zero elevation 0 » 0°
9 » 4°

TRUNNION RIACTIONS.

0*19°

750

Y - F sin(0+6)+Wg CoS 6

fl K

x « F cos (0+e(i- -SO+K -*.r fi sin e

Mr mr

K »(Wr sin 6+B) - 247000 Ibs.
K = PmxA -32000*78.54 = 2513000 Ibs.
Wg = 76830 Ibs.

As a check, we may consider the forces external
to the system above the rollers.

F cos(0+9)ht+F sin(0+e)Lt+Wrcos 8Lt-Wrsin ht-[P cos
(0+e)-K]ht-Be * ZM0

2376000 "31 « 73,656,000

2513000 " .3256 * Ib * 12,273,500
89820 x .9976 x 15 = 1,344,000

87,273,500

89820 x .0698 x 31 * 194,000

2376000-247000 x 31 » 66,000,000
240700 x 12 « 2,889,000

69,083,000

ZM0 « 18,190,500 moment of the rollers
ZN=F sin(0+6)+Wrcos 6 = 2513000* . 3256+89820* .9976

» 907600 Ibs. total normal
load on the

7fift^0

X * 2376000(1- 2)-76830x. 0698^247000

343570-5.363 + 211284 = 549500 Ibs.

751

Y = 818000+77000 » 895000 Ibs.

5Z5.000

in

C\J

GUN

Sectional Modulus=195

Force on the trunnions *

* 10s X30.2+80.1

* 10« /1. 103

= 1.050000 Ibs.

525000*. 3. 375

S = - = 9080 Ibs/sq.in. fibre stress
195

ROLLER RBACTION.

about front roller,

F cos0-K

752

B » K-Wrsin9 » B » 240700

2M(o)» Xht+YLt+Wccos 9 Lt-Wcsin 9 nt-«c ^f bt- Be

d'x
dt«

549500x31+895000xl5+13000x.9976xl5-13000x

.0698*31 - (2376000 - 247000) 130°° x 31 -

89820
240700x12 * 18185000 moment on

V+Wccos 9

the rollers.

895000+13000X.9976 - 907960 Ibs.

total normal
load on rollers

jigACTIOM OK TH8 TRAVgRSING BOLLIR8.

F sin(0-e)+WtWs-Ksin 9

(P cos 0-K)cos 6 +F sin (£l-«-6)

wt. of reooiling part 89000

Mt. of the rest 53000

2513000X. 2588+89000+53000 (-247000). 0698

650000+89000+53000-17200 = 774,800

(2513000x9455-247000). 9976+2513000X. 2588

2124000 + 650000 * 2774000

667,600
53,000

AO"O^V ,- I

r^r ~r~r T L

VV )jh L i4 ln

20 rollers

117,000"

753

SECTION MODULUS
537,

50,000

M(0)-722000x80+53000x65 » 61450000 inch/lbs.

moment on the rollers.

M «K(2 1J+2 If lg)+Kc(2 lt+21t 2 ln-x+1,,)

V « XN-K(2 lt+2 lf 2 ln_1+ln)+2Kc

6145000»K(2x2"73.+2x974 +2x2072 +2x34.4 +2x49,. 5

+2x64.9 +2x78.8 +2x89.6 +2x96.4 +99 )+Kc
(2x2.3+2x9.4+2x20.2+2x34.4+2x49.3+2x78.8

+2x89.6+2x96.4+99)
hence 6145000=73600K+990Kc

774800" 990K+ 20Kc

61430000
34000000

73600K+990KC
49000K+990Ko

27450000 » 24600K

Hence 8 - 1180

Reaction on the last roller

99x1180=117000 Ibs.

Force due to rifling and its effect on the travers-

ing chain.

Frt = Iw

rt

MK*n

rt

2 M

754

v » 5 in.
t - .0162
« . 60S

32.2

R = .8r = 4 rifling 1 turn in c5 caliber
1 turn in 250 inches * 20.83 ft.

" * 2 "

606x4*x770x2

77° ""••

606x16x770

32. 2x5x. 0162x12
238500 Ibs.

32. 2x5x. 0162*12

7465920

-

312984

Torque » 238500x4 » 95400 ft. Ibs,

95400x sin 15° * 9540Qx.2588 = 24700 ft. Ibs.
" TR sin 0 = 24700

TD « — x 24700 « - 2500 Ibs.
49

Tension on the chain at pinion

2500

500 Ibs.

*

.

755

VELOCITY OF FHCC HKOL

mnfi offnojfcn.f-X3.7i

TfVnfLfD BY RECOLIH(,f*KTf DURING

z.ssr re.

KccotLiNs mxrs

OfPHOJECr/tt

/O-INCH BARBETTE CAfff?/AG£

MODEL OF /893
THE EFFECT OF. THE TRAVEL OF

THE PROJECT/LE UP THE
BGHE ON THE ELEVAT/MG ARC

CHAPTER XII.
DOUBLE RECOIL SYSTEM.

OBJECT In order to reduce the reaction of recoil
on a carriage to a moderate value when the
caliber is large a long recoil is necessary.
A long recoil requires long guides and in
addition is usually prohibitive due to
breech clearance necessary to avoid a great loss
in stability due to the overhanging of the recoil-
ing weights at low elevation when the gun is out
of battery, etc. A long recoil may be avoided by
the use of a double recoil system and the stability
of a railway or a caterpillar carriage at the same
time increased. This latter factor is the real
distinctive value of a double recoil system over
a corresponding single recoil system.

It is important to note that a caterpillar
or railway car braked with a single gun recoil
system is essentially a double recoil system, the
ground or rail offering a tangential reaction which
corresponds to the reaction of the lower recoil
system.

Obviously when a top carriage moves up an
inclined plane under the recoil reaction of the
gun and the resistance of the lower recoil system
or when with a single recoil system the cater-
pillar or railway car runs back on the ground or
rail under the recoil reaction of the gun and
the resistance of the ground or rail, the recoil
reaction of the gun becomes different and the
throttling grooves must therefore he necessarily
different, then with a single recoil system when
a constant recoil reaction is imposed between the
gun and top carriage.

757

758

CLASSIFICATION. In the design of a double recoil
system it is desirable in order to
simplify calculation and secure
uniformity of stresses throughout
the recoil to have both the upper
and lower recoil reactions constant throughout
recoil. However, in ordnance design it has been
customary to mount single recoil mounts, gun and
top carriage together on caterpillars, etc., and
for augmenting the stability to allow the top
carriage to recoil as well up an incline plane,
the inclination of the plane being sufficient to
bring the systeu into battery after the recoil.
The recoil reaction of the upper system can there-
fore, with a double recoil no longer be constant
since the recoil reaction is the sum of the air
reaction, a function of the relative displacement
between the gun and top carriage, and the throttling
reaction which is a function of the relative velocity.
Therefore, with a constant braking on the lower re-
coil system, to ascertain the displacement of the
top carriage up the incline plane, it would be
necessary to carry on a somewhat elaborate point
by point integration for the various dynamical
equations and displacements at each point of the
recoil.

Hence in the following discussion we will con-

•

sider the dynamical relational-
CD With a constant resistance for

both upper and lower recoil systems.
(2) With a given upper recoil system
and a constant resistance for the
lower recoil system.

APPROXIMATE THEORY FOR (1). Reactions and

velocity for double
recoil systems: Let

P = resistance of gun recoil system
W or wr * wt. of recoiling parts (upper)

759

O/V

SYSTEM

Fig.l

760

WgOr wc = wt. of top carriage and cradle (lower)

V = initial velocity

Z * displacement of gun on carriage, i. e. =
relative displacement

N = upper normal reaction between recoiling
parts and top carriage

M - lower normal reaction between top car-
riage and inclined plane.

X = total run up on inclined plane.
/ or v = velocity of combined recoil

t = corresponding time for combined recoil

0 = angle of elevation of gun

6 = inclination of inclined plane.

Since during tbe powder pressure period, there
is no appreciable movement of the top carriage up
the inclined plane, and the timeaction of both
the upper and lower recoil reactions is negligible
as compared nith their time actions in tbe pure
recoil period after the ponder period, we may as-
sume the recoiling mass to have an initial velocity
V at the beginning of the recoil, where

wv0+« 4700

V = 0.9 ( )

"r

where w = weight of projectile
w = weight of charge
v0 = muzzle velocity

and 0.9 is a constant to allow for the effect of
the recoil reaction on the recoiling mass during
the powder pressure period. Consider now fig.(l)
Tbe retardation of the recoiling parts is the
vector difference of tbe velocities at the end
and beginning of tine t divided by "t", that is

v-V
a - hence assuming axes parallel and normal

to the guides of the upper recoiling
parts, we have the following equations of motion
for the recoiling parts,

761

g

snd « ,_ ^x
^ wr v sin(9+0)
N-Wrcos0 = -5 '— (2)

g t

Since tliere is no roation the cou pie between the
recoiling parts and top carriage need not to be
considered.

Next considering the motions of the carriage
above, we have, along the inclined plane:

H,

„ i „ O D _ .

C1

P cos(0+9)-N sin(0+6)-Wcsin 6 - R = — - (3)

and normal to the inclined plane,
N cos(0+6)+Wccos 9 +p sin(0+9)-M = 0 (4)
If, after the recoiling mass and top carriage
are brought to a common velocity, we consider
both as a single mass in motion neglecting the
effect of the further motion of the gun on its
slide, the common mass

brought to rest by a constant force H.

g

Hence the retardation after time t, becomes,

Rg
ar = -j — *r and the interval of common retardation,

r c becomes, ^ +^

tr = and the corres-

yj +w "g ponding displace-
t. r c
ment - artr= v* • Therefore, the total dis-

"g placement (since the top
carriage is uniformly accelerated to a velocity
v at time t)becomes.

Since the relative displace-
ment equals the absolute
displacement of the gun parallel to the guides
minus the displacement of the top carriages
parallel to the guides, we have

V+v cos (0*9) v cos (0+9) V

Z = t t hence Z * - t

22 2

762

763

ENSRGY SQUATION FOR Let x1 and y1 = the co-
DOUBLE RECOIL. ordinates parallel and nornial

to the gun axis.

x and y = the coordinates parallel and normal to
the top carriage inclined plane.

v

xt = -j- t where xx = the displacement of the top

carriage up the inclined plane
at the instant when the re-
coiling mass and top carriage
icove at common velocity v,

hence xt = - t . Then for tne recoiling parts, we

have, (P-Wr sin0)x'= i [ V*-v2cos2 (0+6 )] (!') (In

direction of upper guides) and

(N-Wrcos)xtsin(0+9) = ^rorv2sin2(0+6) (21) (at

right angles to upper guides), and for the top
carriage alone, we have

[P cos(0+9)-N sin(0+6)-Wcsin 6-R]xt = \ Mcv2 (3 ' )
(Top carriage up plane)

Subtracting (31) fros (I1), we have
P[xl-xtcos(gJ+e)]-wpsinefx' + N sin(0+6)xt+Wcsin 8.xt

+Rx« in.fV'-v^os8 (0+6)]- - 0 v2

S f 2 t»

and substituting (2) in (4), we have
P[x'-xtcos(0+9)-Wrsin0.x'+ ^mrvasin2 (0+6) +Wrxtcos
0sin(0+Q)- ~ mrtV*-vacos2 (0+6)]+ i mcv2 + wc sin

9.x+R xt = 0
Now the relative displacement between the gun and

top carriage becomes, Z = x'-xt cos (0+6). Hence

the above expression reduces to

PZ-Wr[x' sin0-xtcos0sin(0+6)]+ |mrva+ %cv*+Wc sin

e.x+R xt = | mrva (7)

Now Wr[x 'sin0-xtcos0sin (0+6)] is evidently fhe
work done by gravity on the recoiling parts and
Wc sin 9.x is the same for the top carriage. In
terms of the relative displacement Z, the work done
by gravity on the recoiling parts may be obtained
by consideration of fig. (2)

From fig.( 2), we have, x '»Z+xtcos (0+6)

764

and the work done by gravity on the recoiling

parts becomes, Wp(Zsin0-xtsin 6)

= Wrtx'-xtcos (0+9)sin0-xt sin 6]

*Wr(x 'sin0-xtsin0cos 0cos 9+xtsin0sin 9-x sin 6)

* Wrt* 'sin0-x^sin0cos0cos 6+x sin 6 (sin*0-l]

= Wr (x 'sin0-xtsin0cos 0-xtsin 6 cos2 0)

= tfr(x'sin0-xxsin(0+9)cosen

Hence equation (7) reduces to,

PZ-Wr(Zsin0-xtsin 9)+Wrxtsin 9+i(mr-nn

v
where xt = - t . Further since R(X-XI)= 7(<nr+mc)v2,

equation (8) reduces to

PZ-Wr(Zsin0-xsin9)+Wc x sin 6 +RX = J mrva (9)
Equation (8) is almost obvious from the theory of
energy, since the total initial energy - mrv* plus

the work done by gravity Wr(Zsin£J-xtsin 9)-Hcxtsin 9
equals the final Kinetic energy of the system
-(mr+mc)v* plus the work done in the upper and
lower recoil brakes PZ + Rxt to the combined re-
coil.

Equation (9) is also self evident since the
final Kinetic energy of the system equals zero after
the system has recoiled the total displacement x
up the inclined plane.

RECAPITULATION OF APPROXIMATE When the re-
FOJRMULAE fOR DOUBLE RECOIL sistance to recoil
BRAKES. is assumed constant

for both upper and
lower recoil systems

we nay with a very close approximation obtain the
principle reaction, by the previous derived formulae.

These formulae are recapitulated in the following
group for convenience in calculation. Then if,
w = weight of projectile
w = weight of charge
v0 » muzzle velocity

765

Wr or

wr * weight or recoiling parts
w0 = weight of top carriage
Z = displacement of gun on carriage
X = total run up on inclined plane
If = upper normal reaction between parts.
0 = angle of elevation of gun
9 = inclination of inclined plane
V * initial velocity recoiling parts
v = velocity of combined recoil
t = corresponding time for combined recoil
x = run up on inclined plane to combined recoil

We have wv_+w4700

V = 0.9 (— ) (!•)

wr

wr ,V-vcos(0+9),
P-wrsin0 = — [ ] (21)

« „ "r v sin(0+9)

N - wr cos 0 = — s (31)

g t w
P cos(0+9)-Nsin(0+9)-wrsin 9-R = — - (4')

g t
v *r+we

••st'-sT1" -\ «**»*.$&« (6>>

_ v

- 2 ^6 '

Usually Z and x are given. Hence, the unknowns
are V, P, t, v, K and R; therefore a complete
solution is possible.

A final check may be made by substitution in
the energy equation:

PZ -wr(Z sin0-x sin 9)+wcx sin 9 + i(mr+mc)v*+Rxi
= ^mrVa (71)

where xt = - t or in the form
PZ-wr(Zsin0-x sin 9)+wcx sin 9 +Rx = 7"irV*

In a preliminary layout for a double recoil
system, the limitations are usually the length
of upper recoil, that is the total relative dis-

766

placement between the gun and top carriage, and
the total run up the inclined plane. A direct
solution of the various reactions in terms of
these given quantities is especially useful.
a a

h * cos(0+9)

b = M
g

1 = sin(0+6)

nrcos(0+9)

c = -

g

n = i»r sin 9
d * nrcos2f

"c

g * —
g

wrsin(0+6)
f « -
g

2«

2Z

(6) t = — same as (5) gives t directly

b v

(7) p » a+- - c - f(p.V) same as (1)

t t

(8) N = d + f - f(N.V) same as (2)

t

(9) hp - IN- N-R - g- f(p.N.R.V.) same as (3)

(10) X > — + p — f(R.V) same as (4)
2 R

Elimination N

(11) hp - Id - i^ v - n - f v = R (8) in (9)

t t»

767

f (P.R.7)

pV2 ' —

i- 5 v from 10 f (R.V. )

o

Elimination R

(12)

If+g
(13) P , -- *-- v * -_ 2 n

xh- —v

2 f(P.V)

elimination of P

b cv ld+n Id + g

~ • - • *

""-'

,,_x aht bhx bhV chx p ch

(15) ahx v~t 2 T T" ~ x<dl+n)

t(dl+n)y x fl+g

+ (fl+g)V+ £-V2-pV*=0

2

ch + fl*g _ aht b_h chx _ t(ld-«-n) ^ x(lf-t-g)

2 2 2 2 t 2 t

bhx
+ ahx + - x(ld+n) = 0

This equation may be put in the form a1 V2+b ' V+c '=0
w_cos2(0+9) wr+Wr wrsina(0+6) WP

i C w

a1 = + + —

2g 2g 2g 2g

wr Wr+Wc Wc
or a1 = — - + — hence a' = 0

2g 2g 2g

Solution of b '

aht Z " t

+ = w sin 0 cos (0 + 6) - = sin(? cos(0+8)

2 V V

768

bh *rv "rv

2g

"r
003(0+8) = r^- cos (0+6)

^

* - " - ;TT - * TT" cos2 (0+6)

a 2gZ 2gZ

0Qt wrcos0sin(0+6) 2Z»r

- - = ~2Z - — - = — — cos0sin(0+9)

2V V

nt

sin 9
V v

xg _ xVwc
t 2gZ

hence xVwr Zwr wrV WCZ xVwc

- sin9 + — cos (0+9) sin9+ -

2gZ V 2Wg V 2gZ

simplifying

xV Z w V

(wr+Wp) sine (wr+Wc)+-c-cos (0+9) =» b1

2gZ V 2g

2g

+ ahx + x*rsin0cos (0+9)

bhx wr^
+ = x — — cos (0+9)

t 2gZ

- dlx = - x w cos 0sin(0+9)

- xn * - x Wcsin 9

xwrVa

xwP[sin0cos(0+9)- cos0sin(0+9)]+ cos(0+9)-xW.sin9

2gZ

- xwr[8in(0+9)cos0-cos(0+e)sin0]

- xw.sin 9 + cos (0+9)- x W.sin 9

2gZ

769

xwrV2
C1 = — — cos(0+9) - x sin 9(wr+Wc)

..V2

cos(0+9)- x sin(wr+Wc)

(wr+W,J(— - - sin 9)+ cos(0+e)

r c 2gZ V 2g

As an example of the solution of these equations
and a calculation of the prime reactions, the 240
m/m Schneider Howitzer was taken with the top car-
riage moving up a plane inclined at 6° with the
horizontal and with 40" upper recoil and a total
of 30" recoil up the inclined plane for the lower
recoil.

Muzzle velocity Vn 1700ft/sec,

Travel up the plane x 30in.

Length of Recoil L 40 in»

Angle of elevation 0 20°

Angle of plane 9 6°

Weight of carriage Wc 11,500 Ibs,

Weight of gun wr 15,800 Ibs,

Weight of the charge W 35 Ibs.

Weight of projectile w 356 Ibs.

Relative displacement Z=L- 40 in.

wVm+4700 w ^356+1700+4700+35.

V « 0.9( ) = 0.9( )

wr 15800

„ 6070+1640 7710

= 0.9 — — = .9 = 44 ft/sec.

158 158

sin(0+9)=sin 26° = .4384
cos(0+9)=cos 26°=. 981
sin 9 = sin 6° = .1045
sin 0 = sin 20° = .342
cos 0 = cos 20° = .9397

770

2,22 .1045). , .891

44 64.4

= 27300C.5124 -,0081)+9620
» 13770+9620
b1 = 23,390

,
64.4x40

.891 - 2.5 x 27300 x .1045

- 317406 - 7132
c1 = 310,274

23390 V = 310274

2Z 2x3.33

t = — = — — — = .1515 from (5)
V 44

13.2652 27300 2

2.5 » — x .1515 + -— — — - from (4)

64. 4R 13-2652
74459.408

R

74459.408
R = = 49800

1.4952

13.2652x.4384

N » 15800 x .891 + 490 x — — - from (2)

• 1 bib

= 14080 + 18810
N = 32890

P - 15800x,342 + 490 44~13'2652x -891 from

.1515

5404+490x212.
P - 109,500
t - 0.1515 ft/sec.
V » 13.2652
R » 49,800
N = 33,000 P = 109,500

771

As a final check on the calculations, the
values obtained were substituted in the energy
equation. The slight discrepancy between the
two sides of the equation is due to numerical
approximation.

v8 i

PZ+ — (Mr+Mc;+x .Wcsin e +x'R= -MrV+wr (Zsin0-x 'sine )

Ct

x1 = t = 1.0048

365000+74, 5000+1200+50, 000= 490, 700
475,000+16,400 = 491,400

dev. 1.42X

EXACT THEORY FOR CONSTANT Let x1 and y1 = the
RESISTANCE ON BOTH UPPER coordinates along and
AND LOWER RECOIL SYSTEMS. normal to the axis of

the bore (upper recoil
coordinates ).
x and y = the coordinates along and normal to the

inclined plane (lower recoil coordinates)
mr and »r = mass and weight of recoiling parts

mc and wc - mass and weight of top carriage plus

cradle

v = velocity of any instant along inclined plane.
?' = absolute velocity of recoiling mass along axis

of bore.

t = time from beginning of recoil
0 = angle of elevation of gun

9 = angle of plane

E = free recoil displacement for upper recoiling
parts during powder period

T = total time of powder pressure period
p = resistance of gun recoil system

772

N = upper normal reaction between recoiling parts

and top carriage
R - lower recoil resistance parallel to inclined

plane

n = the coefficient of sliding friction
pb = total powder pressure on breech at instant t,
Hence t1 = time of common recoil

v = common recoil velocity for both recoil-

ing parts.

xj = absolute displacement in the direction
of the bore to where the recoiling
masses move with common velocity
x = corresponding displacement up inclined

plane at common velocity
Z = total relative displacement between

upper and lower recoiling mass.
Vf = free velocity of recoil (See "Dynamics

of Recoil")
RlJ = counter recoil buffer resistance for

upper recoil system.

Considering now, the motion of the upper recoil-
ing parts, we have ,

Pb-P+wrsin0 =mr - (1)
dt

dv
N = wrcos 0=mr — sin (0+8)- (2)

From (1), we have

that is,

t Pfcdt (p-wrsin6)
/ --- t

(p-w_sin0)
V. -- - - t = v' (3)

mr

now, when t = T V^= Vj • hence at any time after
T, we have

(p-wrsin0)
Vf i -- t = v '

773

Integrating again for the upper recoil absolute
displacement, we have

t (p-w_sin0)
Vfdt -

0 2mr

but tit

/ Vfdt = / Vfdt + / Vfidt
o o T

Hence (P-w_sin0)

x' = E + Vft(t-T) t» (4)

2»r

which gives the absolute displacement of the upper
recoiling parts along the axis of the bore.

Considering now, the motion of the lower re-
coiling parts, ne have,

p cos (0+6) - JT sin (0+8) - wcsin e-R»« — (5)

' dt

Substituting N from equation (2) into (5) and
simplifying, we have

p cos(0+9>-wrsin(0+9)cos0-wcsin 9-R=[flic-mrsin*00+e)~

dt

(6)

p cos (0+9 )-wrsin (0+9 )cos0-wcsin9-R

Hence v = [ ] t (7)

mc-mrsin*(0+9)

and the corresponding displacement up the plane,
becomes

pcos(0+9)-wrsin(0+9)cos0-wcsin0-R
x - t = ] t. (8)

2[mc-mrsin»(0+9)]

The relative velocity between the upper and
lower recoiling parts, become vr=v'-v cos(0+9) (9)
and the corresponding relative displacement
xp=x'-x cos (0+9) (10)

When the upper and lower recoiling mass move to-
gether with a common velocity, vr = 0, hence
v'«v cos (0+9) hence we obtain the time t1 for the

774

comaon velocity, from

p-w_sin0 pcos(0+9)-w_sin(0+9)cos0-wftsin9-B
?ff , ( - £ - ) t* [

nc-«rsin*(0+9)

cos (0+9) t1
simplified, we have

p-wrsin0 Pcos(0+9)-wrsin(09)cos0-wr3in9-R
«r mc-mrsin»(0+9)

(12)

cos (0+9)

As a check, the time t1 for attaining the common
velocity of the upper and loner recoil masses,
we may equate the components of the absolute
velocities of the upper and loner recoiling mass
parallel to the inclined plane.

Considering the motion of the recoiling parts
parallel to the inclined plane, we have

v » Vf • cos(0+6) - £- cos(0+6)t'+ — sin(0+8)tf
«r »r

since now the reaction N has a component N
reacting on the upper recoiling parts parallel to
the inclined plane.

Let Nt* = wrcos 0 t1 = •rsin(0+9) v hence

D "r

+9)- -2- cos(0+6)t'+ — sin(0+6 )cos0t '+sin*

•r "r

"r
(ef+9) v -- sin 6 = v

•

0+e) = Vf,(0+6)- S- cos(0+6) t' + ~
mr mp

cos0-sin 9] t
Let sin(0+9) cos(?-sin 6=(sin0cos0+cose)cos0-sin9

775

* sin£fcos0cos9 + cos* 0 sin 9 - sin 9

= sin2fcos0cos6 + sin e (cos'0-l)

= sin0cos0 cos9 - sin 9 sin* 0

= sin0cos(0+9)

bence p-wrsin0

v cos(0+9)=V£i - ( )t '

r

Substituting for v and reducing, we have, as before

t, _ !£J _

p-wrsin0 p6os (0+9 )-wrsin (0+9 )-wcsin8-R

. -,.. .
Br mc-fflrsin«(0+9)

Knowing the value of t' and substituting in equations
(4) and (8), we obtain the total relative displace-
ment between the upper and lower recoiling parts,
that is Z = xj- xtcos(0+9) (13)

where t1 is used in the values of x1 and x res-
pectively.

The total energy of the system where the two
masses arrive at the common velocity vt, becomes

Z Z

-(rar+Rc)v* + / Pa^Z where / padg is the potential

energy in the

recuperator. Let R^ = the buffer resistance during

counter recoil for the upper
recoil system, then

Z
/ RjJ d Z = the work done by the buffer in the

upper recoil system.
If now we assume that the counter recoil of the

upper system is completed during the recoil of the
lower system, we have

Z Z

i(mr+mc)v; + / p d Z = B(X-xt)+ / R,J d Z

= wr(X-xt)sin0+wrZsin0+wc(X-xi)sin 9 (14)
A physical meaning and relationship of the re-
actions in this equation, may be had oy a con-
sideration of the component dynamical equations
for the parts' of the system.

776

Daring this second period of the recoil,
we have for the lower recoiling mass, that
[R+wcsine+Nsin(0+e)-(pa-Rb')cos(0+e)]dx = - ncdV
and for the upper recoiling parts, along and
normal to the boref (pa~RjJ )-wrsin0)d(x cos(0+9)+Z]
~mrvxdvx and (wreos0-H)d[x sin(0+e)}*-mrvjdvj
and the above equations and integrating the sum,
ire have

X X

/ R dx

»a

+ / wrsin6 dx + / (pa-Rb« )dZ+wr f [sin
* t x

(0+6) cos 0 - cos (0 + 6)sin0] dx
o v«

- / wrsin0 dZ=(mr+mc)r^ (since v£*+ v^2= v* for

initial valve)
Simplifying we obtain equation (14)

In general the potential energy of the re-
cuperator is partially divided in overcoming the
work of the upper recoil buffer

Z
/ R£ dZ and in augmenting the run up the inclined

plane over that if there were no re-
cuperator present. Hence in general

Z Z ^ 2

/ padZ > / Rfa' dZ and / p. dZ - / RJ dZ
o o o o

is the additional 2nergy over that Kinetic energy
at common velocity which augments the recoil up
the inclined plane.

We nay assume with small error, however, that

Z Z Z Z

/ pa dZ - / RjJ dZ or that / pa dZ - / R^ dZ is

negligible. This does not imply that pa-R£= 0.
Since (pa~R^ ) (X-xt) cos (0+6) (roughly) is the

agent by which the upper recoil energy is dissipated,
When the lower recoil is comparatively short

777

and the resistance of the lower recoil system R
is large, we have often a condition, where
counter recoil in the upper recoil system be-
comes impossible and we even have an over run of
the upper recoil system.

Thus assuming during the second part of
double recoil, that the upper and lower recoil
mass move as if one, we have the retardation

dv R+ (wr+wc)sine

- — - * -——_——— and for the upper recoil-
df mr+mc

ing mass

dv

Pa- wr sin 0= mr( -- )eos(0+e) hence
dt

m

Pa-wrsin0 = [R+(wr+wc)sin9]cos(0+9)

mr+mc

Now if pa > wr sin 0 + — — [R + (wr-rw_)sin0]cos(0+9)

mp+mc

Counter recoil of the upper recoil system is pos-
sible during the second period of the lower recoil
system. If however,

mr
Pa < wrsin0+ [R+(wr+wc)sin 9]eos(0+6)

We have a tendency of over recoil of the upper re-
coil system hence counter recoil of the upper re-
coil system is impossible. For this case the energy
equation reduces with exactress to

i(mr+mc)v«*H(X-xt)+(*r+wc)(I-xt)sin 6 (15)

The velocity curve during the second period
may be obtained with sufficient exactness by as-
suming the two masses to recoil together, then

<*»
R+(wr+wc)sin8a-(mr+mc)v —

x v

and / [R+(wr+wc)sine)dJc » ) (mr+rac)v dv

778

mr+mc
[R+(wr+wc)sin6] (x-xt) = (-— )(va-v*

hence /~ a[8*(.r*n0)«ine]U-«t)

?.»/.?• '

mr+mc

, RECAPITULATION OP FORMULAE Prom approximate
FOR CONSTANT RESISTANCE TO solution with limited
RECOIL BOTH UPPER AND upper and lower re-
LOWER. coil, calculate

P and R. Then during
the powder period, we have

p-wrsin9

v' . Vf ~ ( ) t

rar

pcos(0+9)-wrsin(0+6)co30-wcsin9-R
mc-mrsin2(0+9)

and the relative velocity becomes v^v'-v cos(0+9)

t p-w_sin0
x1 = / Vf idt - ( )t«

o 2mr

pcos(0+9)-wrsin(0+9)cos£l-wcsin 6-R

and the corresponding relative displacement, be-
comes xr=x ' - x cos(0+6). After the powder period

during the remainder of the first period of recoil,

we have p-wrsine!

v'=V i( - )t

3 t

and for the relative velocity vr»v'-v cos(£l+9)

Furtber p-w.sinar

x'«E+Vf i(t-T) - ( )tf

2m_

779

p cos(0+9)-w sin(0+9)cos0-w.sin9-R
x .[ - £ - 2 - ]t.

2ac-»rsina(0+9)

and the corresponding relative displacement be-
comes xr»x'-x cos (0+8). The tine for the common
velocity becomes,

Vf,

t'=

p-wpsin0 pcos (0+9 )-wrsin (0+6 )cos0-wcsin9-R

., fl. - ]cos(0+9)

and the common velocity becomes

pcos (0+9 )-Wpsin (0+9 )cos0-wcsin9-R

V**t- mc-H.rsin»(0+9) -U'

p-wpsin0
x'=E+Vf ,(t'-T)-(— - - )t'

-oDDj.

pcos (0+9 )-w_sin (0+9 )cos0-w_sin 6-R
x -f - - - ]ta
2[mc-mrsin« (0+9)]

and the total relative displacement for the upper
recoil system becomes, Z * xj - xtcos(0+9)

Oaring the second period of the recoil, ire
have

v*-2(R+wr+wc)sin 9(x-xt)

v = i

mr+mc

the upper recoiling mass being assumed locked with
the lower recoiling parts.

CALCULATION OF THROTTLING GROOVES As a first
BOTH UPPER AND LOWER RECOIL. approximation,

it will be
assumed that
the total

friction is mainly guide friction and proportional
to the normal reaction between the upper and lower
recoiling parts. Then Rg* nN where n * 0.2 to 0.3

and N » w_cos0 +mp — sin(0+9) where v and t1 have
t'

780

been already determined. Considering the upper

recoiling parts, we have P»pa+pn+Rg*pa+p_+nN
hence pn=p-pa-nN. Further if the ratio of the

final to the initial air pressure = m, we have

Paf

— — « m and if A « effective area of upper

°ai recuperator and b = Z =

recoil displacement on top carriage, then for the

initial volume we have

x

7j * Aab • where k = 1 to 1.41 assume 1.3

mk - 1

vi
Hence pa=Pai( )

vi *axr

xr = being the relative displacement. There-
fore knowing vr and xr and the total pull p, we
have

/ -i \ »» "n * r

PnsP-PaiV-A > and Wx i = —

Vj_k-nN KAnV

i-Aa

13. ay

Ah

where An= the hydraulic piston area and k the
reciprocal of the throttling constant.
Lower throttling grooves

Knowing R from previous data, we have

R A v

where v is the recoil velocity up

13. 2/- plane.

EQUIVALENT MASS OP ROTATING When a double re-
PARTS WITH A DOUBLE RECOIL. coil system, con-
sisting of two
separate recoil
systems is used,

mounted on a railway car or caterpillar, it is
customary to consider the car or caterpillar

781

sufficiently braked to allow no recoil. In fact
a salient feature of the design is to make "R* small
enough so that the rail or ground friction, induced
by proper braking, is sufficient to balance R.

Due to the complication of a double recoil,
as well as the impossibility with very large mounts
of taking up the recoil energy even with a double
recoil system without an excessive recoil displace-
ment it has been the custom to use a single recoil
and allow the railway car or caterpillar to run
back a limited distance dependent on the magnitude
of the braking. The recoil of the car on very large
railway mounts may be considerable. This greatly
reduces the stresses on low elevation as well as
augments the stability. In fact with such mounts
stability is of no longer a consideration.

When a single recoil system is used but the
car or caterpillar recoils in addition, we obvious-
ly have a double recoil system and all the prevous
dynamical equations together with the method of
computing the throttling on the upper recoil or
now the recoil systems holds the same. The
lower recoil resistance R is now the tangential
reaction exerted at the base of the car wheels or
at contact of ground and caterpillar track. In
the acceleration of a railway train, railway engineers
customarily allow for the rotational inertia of the
car wheels by increasing the translatory mass from
8 to 10 percent. Due to the limitations at times
of car or caterpillar recoil and the great
variation of the magnitude of the rotational in-
ertia as compared with standard railway practice
it is important to calculate the exact effect of
the rotational inertia in terms of an equivalent
addition to the translatory mass.

Consider a railway car or truck with "n"
pairs of axles. Let

wc and mc = weight and mass of car not in-

782

eluding wheels.

•„ and BW * weight and mass of a pair of wheels.
I * D*K* = moment of inertia of a pair of

wheels about the center line of the

axle.
d * tread dies. of a car wheel

k * radius of gyration of a pair of car wheels
Nw = normal reaction at base of a pair of

car wheel x
Kg * normal reaction of brake shoe on wheel

per pair of wheels
fw « coefficient of rail friction
fa * coefficient of brake shoe friction
R,, * tangential force exerted by rail oc base

of car wheel
p * recoil reaction
N * normal reaction between recoiling parts

and car

0 * angle of elevation

Now independent of rotation or any other
motion, the translatory motion of the center of
gravity of a system depends only on the external
forces applied. Hence

p cos 0 - N sin £> - 2 Rw « (mc+Znw) —

dt
Considering the motion of a single car wheel, we

have for rotations about the center of gravity
of a pair of wheels

hence pcos0-Nsin0-2N8f s=(mc*2mlf+Z — - — ;— there-
fore the translatory mass is increased by the

4mk«

which is the equivalent translatory

783

mass of rotational inertia. The a ass of the
lower recoil system therefore, becomes

4k*

m-+2m_(l+— -—) and this value is to be sub-
0 d

stituted in the previous dynamic

equations. The equivalent resistance for R is
now the summation of the brake shoe friction,- that
is R = Z Ngfg and this value is to be used in place
of R in the previous dynamic equations.

It is important to note, however, that the
actual tangential force exerted at the base of the

* : .. •

car wheels is not 2Nsfs

but ZR, - 2Ngfs + 2 — JT- -

943 '-is '-'-si* • ->riJ-afr HoJ«H»i srff
Consider a caterpillar track and connector

mechanism:

Let Rt» total tangential track reaction between

track and ground (in Ibs)
Rw = total tangential roller reaction on

track (in Ibs)

rw = radius of roller wheel (in ft)
rc * radius of sprocket (in ft)
rt * radius of sprocket gear (in ft)
r = radius of brake drum gear (in ft)
r = radius of drum of brake clutch (in ft)
R * tangential reaction between sprocket

gear and drum gear (in ft)
TQ = torque exerted on brake drum (Ib.ft)
E0 = mechanical efficiency of sprocket

mechanism
Et * mechanical efficiency of transmission

between sprocket gear and drum gear.
Efi - mechanical efficiency of brake drum

and mechanism

EN 3 mechanical efficiency of roller trucks.
A = resultant normal bearing reaction of

sprocket shaft (Ibs)
J* » resultant normal bearing reaction of

784

brake drum shaft (Ibs)

ft and ff « corresponding coefficients of friction

BC = total mass of caterpillar excluding recoiling

parts

mwk£ = moment of inertia of roller wheel (ft. Ibs)
msk| = moment of inertia of sprocket wheel
mgsk§s * moment of inertia of sprocket gear
= moment of inertia of drum gear
1 Moment of inertia of drum,
d * the increment change in the radius to account
for friction between gear teeth

Considering the motion of the caterpillar,

we have

dv

p cos 0-N sin 0 -Rt * mc 77

The tension in the caterpillar track at the
sprocket becomes, T = Rt-2Rw

. R *"k" d* (2)
Bwr* dt

and its moment about the sprocket axis, (assum-
ing the upper track tension as nil) becomes,

(Rt

mwkw

« dt

Considering the angular motion of the sprocket shaft
we have

mwkw dv , mskg-nng3Kgs dy

'

(3)

Further considering the angular motion of the drum
shaft, we have

dv

(0

hence

si, „,

785

Where E^ takes care of the friction less in the
drum of gear bearing.

The friction loss between the drum gear and
sprocket gear may be considered, by letting

Illl * 1 li
r,-d 3 E r,

hence

Substituting in (3), we have

v£ dv TQ ix

Rt » ^o-

EBr» dtoE EEErr»t dt

where EO takes care of the friction loss in the
sprocket bearing. Therefore, the track re-
action becomes,

k| dy

(8)

and substituting in the equation of translatory
motion (Eq.l) we have

TD rt mHk«

p cos 0- N sin0 — — = [mc+ '."" +

BoM. r.r0 EBrS

^

TD

Evidently — — — — is the brake torque referred
EoEtE2r2ro to a reaction at the base of
the track, considering the mechanical efficiency
of the gearing . The translatory mass is augmented

786

due to the rotational inertia of the rotating
parts by the terra

2 msks*mgsk|s

which is the equivalent mass of the rotating ele-
ments.

It is to be noted that the mechanical ef-
ficiency enters in the rotational inertia since
the bearing reactions depend upon the external
reactions, and the moments of them in turn de-
pend upon the rotational as well as translatory
inertia. The effect of the translatory inertia
on the rotating element in modifying the bearing
reactions will be neglected, being small.

Hence R in the double recoil equations is now
the braking torque referred as a tangential force
at the track base, that is,

The actual tr ck re.action is Rt given by
equation (8). As a check on equation (9) ire may

note that from the energy equation, we have

t t

D 9 ' ^dda^nifi^fid ft i
p cos0-N sin0 dx = »-d ( )w' +d(5-racv*)

the reaction Rt doing no work. Further, we have

r, ri v

de* %-r4- dx K = r r

« O a 0

V V

If » W_ =

r " r

ro rw

hence substituting these values, we find

787

T r

. :t.0

therefore the equivalent translatory mass, to ac
count for the rotational inertia becomes,

gdk|d a msks+ mgsk|s
'

When the caterpillar track is heavy or there
is a long space between the driving sprocket
and the front idler sprocket, its inertia ef-
fect must be considered. Therefore, let

r0 = radius of drive gear sprocket (in ft)
r = radius of front idler sprocket (in ft)

t fc » « i i "

m^k£ = moment of inertia of idler sprocket

(units Ib.ft)

mt = mass of caterpillar track per unit
length

1 * length of upper span of caterpillar track
(in ft)

(aromt)r£ = moment of inertia for that part
of track in contact with driving
sprocket (units in Ib. ft)

(nr^mt)r£ = moment of inertia for that part
of track in contact with front
idler sprocket (units Ib. ft.)

T0 = tension at section at point of contact

of lower track and drive sprocket wheel
(in Ibs)
T. = tension at point of contact of lower

3

track and front idler sprocket (in Ibs)
Ta = tension at point of contact of upper

track and idler sprocket
T = tension at point of contact of upper

track and drive sprocket
From kinematics we must have the relative
velocity of the track with respect to the frame

788

dv
equal to v and the corresponding acceleration —

where v is the translatory velocity of the
caterpillar at instant with respect to the
ground.

Considering the lower track since at any
instant it oust be at rest, we have for the dif-
ference of the tensions at its extremities,

Zawkw dv
TO - T, - Rt - j-Jj! g (10)

I*** at

where the second member is the reaction on the
track due to the tangential reaction of the ground

and the reaction of the truck rollers .

Considering the angular notion of the drive

sprocket shaft, we have

(To-Tt)rr-8t<rt+d)-Atftr' •( * n r

(11)
and for the upper track, we have

T - T = 2 ml — (12)

1 * * dt

•ii dv

(Tt-Ts)rt »( — — + n rjnt)— + A1firii-n (13)
ri dt

From equations (4), (5) and (6) in the pre-
ceding discussion, combining with (11), we have,

s?sktfs dv

gs gs ^ n }]dl

dt

(14)

Further equation (13) may be simplified by
considering the mechanical efficiency of the
front idler sprocket mechanism Ej, that is

789

<T.-IK •<-ir' + n r* mt)IT Ei

where E takes care of the bearing friction A^£^

and the loss due to the Deeding of the track at
the sprocket.

Now if we combine (10), (12) and (15) with
(14), we have

Zmwkw , nri ?rox miki c /gd^gd,
D = - + m+ (2— 1+ - + - — )•*— — — E^ + (-?.. 8 .. )r2
* Ewr* ^ Ex E0 rj « Vl *

kj s dv TD rt

E0r§

nri nro

but mt(2 1 + -— + - — )=Mt approx. (total mass of
Bi Eo track)

dv TD ri
B*=Me dt + ^E^ ral.o

Substituting in the equation of translatory motion
we have,

TD ri dv

p cos- N sintf =(Kc*Me)— (17)

Eo£tE2 raro dt

Where Me given in equation (16) is the equivalent
mass that must be added to the translatory mass.

The equivalent inertia may be taken into
consideration more simply by tlie following ap-
proximate method.

The primary rotational system, consists of
the track, the drive sprocket and front idler,
together with the truck rollers.

The reaction of the ground tangentially to
the track, = Rt and the truck roller reaction =

mw'lw dv

~ at

Hence, for tbe primary rotational system, we have,

790

m»kw

r0)— (18)

but considering the rotational system of the drum
and gear, we have

Combining (18) and (19) we have

Hence, we assume the radius of the idler sprocket
and driven sprocket the same, namely, rQ

To account for the loss due to friction, let
the Mechanical efficiency of the greasing re-
ferred to the track, be as follows:

Et = mechanical efficiency of track

EI = J4.E. of front idler

Eo = M.E. of drive sprocket

Et = M.E. of gear transmission

Ef = M.E. of druu shaft

Ew= M.E. of truck rollers
Then equation (20) is modified to:

V,
=U - (21)

tbat " d» TO',
Rt " *c

d t E>n^> _ E_ 1

PRIMARY EXTERNAL REACTIONS WITH With a double
A DOUBLE RECOIL SYSTEM. recoil system, the

791

first period when

the top carriage is
accelerated to a

common velocity for both upper and lower re-
coiling parts and a second period with a re-
tardation for both recoiling masses.

The reactions should be considered during
both periods.
External reactions during first period:

By O'Alembert's principle we may regard the
inertia force as an equilibrating force, then for
the primary external forces of a system con-
sisting of the upper and lower recoiling mass to-
gether with caterpillar or railway car.

(1) The inertia resistance of the
recoiling mass divided into two

components .

(a) The inertia force parallel
to axis of the tore through
the center of gravity of the

upper recoiling parts, p1 or K

xt

(b) The inertia force normal

to the upper guides through the
center of gravity of upper re-
coiling parts, N1 or Kv

*\

(2) The weight of the recoiling mass
acting vertically down = Vr

(3) The inertia resistance of the
top carriage and cradle acting
through the center of gravity of
the top carriage and cradle
parallel to the inclined plane
opposite to the acceleration up the

plane = Kx or mc &— *
dt *

(4) The total weight of the top
carriage and cradle acting vertically

792

down = «c

(5) The reaction of the ground on
the caterpillar track or the re-
action of the rail on. the braked
wheels of a railway mount using a
double recoil, which are divided

into the following components:
(a) The tangential reaction

of ground or rail.
(b ) The normal reaction of
ground or rail which is not
uniform but distributed so as
to produce an upward normal
reaction combined with a couple.
When the mount is just stable as with a light
caterpillar at zero elevation (5) reduces to a
single reaction about which moments are taken and
therefore would not be considered for critical
stability.

The primary external reactions are shown in
fig. (3).

Considering the motion of the upper recoiling
parts, we have, during the powder period,

Pb-P+Wrsin 0 =mr — — for the acceleration

of the upper recoil-
ing parts parallel to the guides. And

A
N - Wrcos0 = mr - - = mr — sin (0+9) for the ac-

dt* dt celeration
of the upper recoiling parts normal to the upper
guides.

The external reaction on the recoiling parts
when considered with the total mount, becomes,

during the powder period,

d2x
P'=KX *Pb-mr - -=P-W-sin£!

r« dt«

parallel to the guides in the direction of Pb.

After the powder period during the first period
of recoil, we have

793

P-Wrsin0=-mr along the guides and H-Wrcos0 -

dt*
dv
mr — sin(0+9) normal to the guides. Therefore

the external forces on the re-
coiling parts during the first period after the
powder period, becomes.

P'=KX =-m reversed =m_ =P-Jf,sin0 along the

1 dt* «• bore and

dv
tf'=Kg =mrI7 sin(0+6) = N-Wrcos0 normal to the bore.

Hence during the first period of recoil either
during or after the powder period, we have
P'=KX =P-Wrsin0 along the bore downward and

1 dv

N' = KV =rar™ sin(0+6) normal to the bore dowaward
* t d t

or = N-Wrcos2f. We have in the above neglected the
powder pressure couple, it being at best small, witb
little or no effect on stability. The inertia
force of the top carriage is evidently

dv

•C"T~ reversed parallel to the inclined plane. Hence

the external forces not including (5) become
(1) pl=Ky =P-Wrsin0 along the bore

1 dv

K' + KV anir-— sin(0+6) normal to the bore
»t dt

(2) Ur= upper recoiling weight, vertically

down,
d v

(3) ""cJ" = inertia force of lower recoiling

parts parallel to inclined plane.

(4) tfc = weight of lower recoiling parts,

vertically down.
The external reactions during the second period:

dv

During the second period, mcT~ = Kx reverses

d t t

in direction since the lower recoiling mass now
becomes retarded. On the assumption that during
this period the upper recoiling parts have the
same notion as the lower recoiling parts, we have

794

STABILITY
COMPUTATION

Y

c* Accf/tration of Carriage up jo/an*

P * Broking Force -f=>-H/,.5">
t • Any/e of C/evo+ion
6 • flng/e of inc/ined P/ane
s about P

r~or any of her point
X'r - TCr -ff'Cos

6) , Yr

i. 3

795

DOUBLE REDDL SYSTEM
STCHAMOND &OMM. HOWITZER (SCHNEIDER,

796

for the inertia force of the upper recoiling parts:

if

mrr— in the direction up the plane and parallel to

the inclined plane. Consideration of the
forces acting when counter recoil for the upper
recoiling parts place during the second period of
recoil will be taken in "variable resistance to
recoil for the upper recoiling parts".
STABILITY FOR DOUBLE RECOIL SYSTEM. Consider-
ing fig. (3)

let ac= **•
c dt
= ac-
celeration of

carriage up plane. N'=niracsin(0+6)

P'=P-Wrsin0 = resistance to recoil for upper re-
coiling parts parallel to guides.

Wr = weight of recoiling parts

*t = weight of caterpillar.

IYC = weight of lower recoiling parts

Let, xryr, xcyc, and x^y^ be the respective co-
ordinates for the variops weights, from the over-
turning point 0, at the end of contact of the
caterpillar track and ground. Then for moments
about 0, for the various external forces, in
battery, we have MQ= (Nleos0+P'sin0+Wr)xr+ (wc+mcac

sin6)xc + (Nlsin0-P'cos0)yr+(mcaccose)yc+wtxfc

and for any other position in the recoil, the
various coordinates of the above equation change
to xr=xr(3lcos0+3cos9), yr=yr-(S 'sinfl-SsinS )

x£ =xc-Scos9, y£=yc+Ssine

•here S'=/ vrel dt, and 3s/ vcdt

vrel = relative velocity between upper and lower

recoiling parts

vc- velocity of carriagrs up inclined plane.
Further let A = Wtxt+WrXr+WcX<!
B=(N'cos0+P'sin0)xr
C =(N'sin0-P'cos0)yr

797

dvc
D = (mc sin 6 ) xc

dt

dvc

E = <mc~ cos 8>yc

F=A+B+C+D+E

For stability, we jnust have F = 0. The critical
position fir stability for the first period is
at the end of the first period when the two re-
coiling parts begin to move with the same
velocity. The coordinates, therefore, become
xr=xr-(Zcos0+xtcose ),y^=yr-(Zsin0-xtsin 9)

xcsxc~xicose> vc=yc+xtsin9

x =dis placement up inclined plane at common velocity
Z=total relative displacement between upper and
lower recoiling mass.

Assuming as before that during the second

period the two recoiling masses move as if one,
we have, for the condition of stability

Mo=Wtxt+wcxc+Wrxr~IDcacsin9 xc" acaccos0 yc"mracsin

P
9 x£ - mraccos 6 y'r =0 where ac=

' mc+mr

and the critical stability is at the end of recoil
and xr=xr-(Zcos0+Xcos9); y ' = yp -(Zsintf-X sin e)

xcaxc~Xcos e> yc=yc+x sine

where X is the total run up the plane.

If, however, the upper recoiling parts move
into battery during the second period while the
top carriage still continues moving up the in-
clined plane, then xr=xr-J[ cos9.

STABILITY WITH A SINGLE RECOIL We have as be-
AND CATERPILLAR BRAKED. fore the same in-
ertia and weight
moments but in
addition rotational

inertia couples. Since the effect of a couple is
entirely independent of the axis about which moments
are taken, we merely have to add in the previous

798

799

moment equation the additional rotational inertia
terms, taking of course account of the algebraic
sign of the inertia couple.

The following inertia couples are intro-
duced with a caterpillar using a simple mechanism
as assumed before:
Stabilizing inertia couples:—

JT* = drive sprocket and bear couple

dv

— = roller truck couples

dt

mik! dv

— — — = front idler couple

r dt

1 r] = track inertia moment where r =

ro+ri

- and 1 = total span of track

2
Overturning inertia couple:-

mdk3+agdk|d dv

( - *~)*.T? = drum shaft inertia couple
r r *d"t

Therefore tfce stability equation

becomes, F = A+8+C+D+E+G+H+I+J+L and for stability
P ^ 0. Where during the first period A=ffrxt+Wrxr

+W.XI , B =(N'cos(2f+PlsinCf)x' C * (N'sin0-P'cos0)y '
H dv

D = (fflc"^ sin e>*c> E = (mc~J7 cos
skgskdv "wkw dv

^ ^

ii dv

mk

J =»[n(r + r»)+2 1 r]

/8ddx dv
- ( - 'rt~"~ where the coordinates

ror» dt refer to point of
contact of ground and track at rear end of track.
During the second period, the 'inertia couples become

800

f?5 ACT/QMS ON 77PP/MG PdfTTS
DOUBLE WML SYSTfM

- fff/KT/OMS ON 71PPM6 f>AffT3 W B^TTfffY—

- ftf ACTIONS ON T/PP/MG P/4ffT3 OVT Of BdTTtffY—

Fig. 6

801

dvc dvc

reversed, therefore, A-D-E-Br— sine. xp-m_ — —

dt dt

cose y_-G-H-I-J-K=0 where —7*- is determined by

dt

the relation,

TDri , dvc
3 (irr-»-mc+me )

where mr=mass of recoiling parts, mc=mass of
caterpillar and mount excluding recoiling parts,
me= equivalent mass for rotational inertia.

ELEVATING ARC AND TRUNNION REACTION In comput-
OF TH£ TIPPING PARTS. ing the various

reactions in a
double recoil
system, we

oust consider the inertia effect of the various
parts in modifying these reactions over their
static values or as would occur with a single re-
coil.

The primary inertia forces induced by the

double recoil are:

For the upper recoiling parts:

(1) The inertia force of the upper
recoiling mass divided into components
through the center of gravity of the
upper recoiling parts, parallel and
normal to the axis of the bore,
respectively.

For the lower recoiling mass:

(2) The inertia force of the top
carriage and cradle acting through
the center of gravity of this
combined mass, opposite to the ac-
celeration, and parallel to the in-
clined plane.

The inertia resistance of the top carriage
and cradle may be divided into two parallel

coapnents through the center of gravity of the
cradle and top carriage respectively the

802

magnitude of the components being proportional
to their respective masses.

xt and yt* coordinates of upper recoiling
parts parallel and normal to
the axis of bore.

x and y - coordinates of lower recoiling
parts parallel and normal to
inclined plane .

Kx = inertia component along bore of upper
recoiling mass through its center
of gravity (Ibs)

Kv = inertia component normal to bore of
i

tipper recoiling mass through its

center of gravity (Ibs)

Kxc = inertia force of cradle through its
center of gravity and parallel to
inclined plane (Ibs)

Kxc = inertia force of top carriage through
its center of gravity and parallel
to inclined plane (Ibs)

Wr = weight of upper recoiling parts (Ibs)
xr and yr = coordinates from trunnions of
center of gravity of upper
recoiling parts in battery,
parallel and normal to bore (ft)
HC = weight of cradle (Ibs)

x- and y« = coordinates from trunnions of
t i

cradle parallel and normal to

bore (ft)
Wc = weight of top carriage (Ibs)

Hc = total weight of lower recoil parts (Ibs)

W -» ~" W A TW

1 *

w"t= weight of tipping parts

X^ and Yt = components of trunnion reactions
parallel and normal to axis of

bore of gun.
£ = elevating arc reaction (Ibs)

•

803

j - elevating arc radius about trunnions

or perpendicular distance to line of

act on of £ (Ibs)
B = total braking between upper and lower

recoiling parts (Ibs)
RI = total friction between upper and

lower recoiling parts (Ibs)
P = total resistance between upper and

lower recoiling parts (Ibs)
N = total normal reaction between upper and

lower recoiling parts (Ifas)
Z = relative displacement of upper recoil-

ing parts wit}) respect to lower recoiling

parts.

Pjj = powder reaction on base of breech (Ibs)
e = distance from P to center of gravity of

upper recoiling parts (in)

Then during the acceleration for the upper re-
coiling parts, we have

d«x
Ph »(B+R -W_sin0)=mr - - along the bore and the

dt*

external reaction on the upper recoiling parts
during the powder acceleration, becomes

d»xt
Kx =Pb-mr g (Ibs) along the bore

1. U L

=B+R-Wrsin0 along the bore

=P-Wrsin0 along the bore

During the retardation,

Br - = -(B+R-tfrsin0) and the external reaction

dt2 on the recoiling parts

parallel to the bore is the inertia force,

d»xt

Ktf - -m- - =B+R-tfPsin0
x* P dt«

= p-wrsin0
Hence either during the acceleration or retardation

804

the external com.ponent parallel to the bore on the
recoiling parts equals the total resistance to
recoil off the upper recoiling parts.. The in-
ertia force normal to the bore, becomes,

t t

Kv 3mr~TT7 (Its) Since - — = v sin(0+9), where

* t at* at

v is the velocity of the lower recoiling parts
up the plane .

Kv = mr & sin(0+6) (ibs)
y» r dt

= N-Wrcosd (Ibs)

For the lower recoiling parts, we have

dv

K«s= -m — (Ibs) along the inclined plane
» *dt

Kxc '-rag --> (Ibs) along the inclined plane

a 2 u t

Elevating arc and trunnion reactions:

Let us now consider the tipping parts, that
is the recoiling parts, together with the cradle.

By the use of D'Alemberts principle the
problem in Kinetics is reduced to one of statics,
provided, we introduce the proper inertia forces..

Further, the nutual reaction between the
upper recoiling parts and the cradle of the
lower recoiling parts:, becomes, an internal force
for the system consisting of the tipping parts.

Therefore, introducing the inertia forces,
we have:

For the reactions of the tipping

parts in battery:-
Along the bore: fig. (6)

t

+Ecose9+Wtsin0-Kxcicos(0+9)-2X=0
dt8

805

Normal to the bore:

Wtcos0-E sine-Kxcisin(9+ef)+Kyi-2Y»0

Moments about the trunnion:
d«xt

(0+9)xc »-Ej - 0 since in the battery position
the center of gravity of the tipping parts is
located at the axis of the trunnions. Further,

t

pu-n- - =B+R-WPsin0*K, i We have, for the
dt2

elevating arc reaction,

Pbe+Kxiyr+Kvixr+Kxci [ycicos(0+6)-xc is in (0+6)]
E= - ' -

-j

and for the components of the trunnion reaction
2X=Kxi+Scos9e+Wtsin0-Kxcicos(0+8)

2X»Wtcos0+Kyi-E sinee-Kxeisin(0+6)

For the reactions of the tipping parts out of
battery:

In any intermediate position, out of
battery the entire tipping parts are displaced
backwards up the inclined plane but in addition
we have a relative displacement between the re-
coiling parts and the cradle of the top carriage,
equal to Z (in).

Therefore, the moment of the tipping parts
about the trunnions, become Wr(lr+Zcos0)+Hcilci=Mt

where lr and lc i are the horizontal coordinates

of the upper recoiling parts and cradle in the
battery position. Since center of gravity of
the tipping parts are located at the trunnion in
the battery position, we have V?rlr+Wcilc i=0
hence Mt=WrZcos0. Then, the reactions along the
bore

806

Kxi+Ecos9e+Wtsin0-Kxcicos(0+9)-2X=0
Horaal to the bore:

Wtcos0+Ky i-Esin6e-Kxc i sin (0+9 )-2Y=0
Moments about tbe trunnion:

K,iyr-»Ky iXr+Kxcicos(9+0)yc,-Kxc.sin(9+0)xc,+Wrxr

cos0- Ej=0

Hence, we have for tbe elevating arc reaction for
a relative displacement Z out of battery

j
and for the components of tbe trunnion reactions,

2X=KX i+Ecos6e+Wtsin0-Kxcicos(6+0)
2Y=Htcos0+Kyi-Esin6e-Kxcisin(6+0)

REACTION BETWEEN UPPER AND In the calculation
LOWER RECOILING PARTS. of guide and clip re-
actions and the bend-
ing stresses in the

cradle it is necessary

to know the nature of the reaction between tbe
upper and lower recoiling parts as well as its
distribution.

The reaction between tbe two recoiling
masses, consists of:

(1) Tbe resultant braking reaction
acting parallel t o the guides
and through the controid of the
various pulls.

(2) The guide friction acting along
the guides.

(3) The normal clip reactions, which
may be divided into:

(a) a normal component per-
pendicular to the axis of tbe

807

our or B/trrf/?y

Fig.7

808

ft&ICTMN B£TW££W UPPfft 4N£> tOWfff fffCOtUMG FMffrS :
POUBL E fffCOfi

F>OS/r/OM

Fig. 5

809

bore,
(b) a couple between the

two parts.

The magnitude of the couple depends upon the
assumed position of the line of action of the
normal component; therefore, we may assume the
normal component in its most convenient position
for calculation. Let

N * total normal reaction between upper and

lower recoiling parts (Ibs)
Nt * front normal clip reaction (Ibs)
Nf * rear normal clip reaction (Ibs)
xt and yt » coordinates of front clip re-
action along and normal to
bore with respect to center of
gravity of upper recoiling parts
(in)
xc and ya * coordinates of rear clip reaction

(in)
M » couple or moment reaction between upper

and lower recoiling parts (inch- Ibs)
Pn*total hydraulic pull including packing

friction (Ibs)
Pa* total recuperator reaction including

packing friction (Ibs)
R * total guide friction (Ibs)
dbs distance from center of gravity of

upper recoiling parts to PD (in)
da* distance from center of gravity of upper

recoiling parts to Ba (in)
dr » distance from center of gravity of

upper recoiling parts to R
n « coefficient of guide friction (0.15 ap-

prox. )

B * 2Pn+2Pa»Total braking (Ibs)
R = n(Nt+Nf)=guids friction (Ibs)
lv * horizontal distance from rear roller

contact of top carriage and inclined

810

plane to line of action of Wr
d = distance from A, normal to lino through

center of gravity of upper recoiling

parts and parallel to bore

Then B db»ZPhdh+ZPada. Considering the re-
actions on tbe recoiling mass in battery, we
have, dtx

Pb-m, * =B+R-Wrsin0, along tbe bore
dt

N=Ky i+Wrcos#, normal to tbe bore
M=Pbe+Bdb+R dr, moments about center of

gravity.

Taking moments about A fig. (7) at the rear
roller contact of top carriage and inclined
plane, we have, for tbe moment of the re-
action exerted by the upper recoiling parts,
on the lower 1

M =B(d-dh)+R(d-d,)+M-N( — ^ +dtan0)

costf

Substituting for M, its value M=Bdb+Rdr+Pbe
and for N, its value N=Wrcos0<Kyi we have

Ma=Pbe+(B+R-Wrsin(?-Kyitan0)d
Ki

r r

COS0

Hence, the reaction of the upper recoiling part

on the lower, during tbe powder period, is equivalent

to:

(1) A couple Pbe

(2) A force through the center of
gravity of the recoiling parts
parallel to the bore: B+R-Wrsin0-Ky itan0

(3) A vertical force through the center
of gravity of the recoiling parts,

Kv.

Wr+ — —
cos0

After the powder period, the reactions on the
recoiling parts, become

811

-m ,J.,~B+R-W,sia(y, along the bore
r dt

N-Kyi+Wrcos0, normal to the bore
M=Bd^+Rdr, moments about the center of gravity
Taking moments about A, fig. (8) at tbe rear
roller contact of top carriage and inclined
plane, we have, for tbe moment of the reaction
exerted by the upper recoiling parts, on tbe
lower, the recoiling parts having a relative
displacement Z. 1 -Zcostf

M*=B(d-dh)+R(d-dr)+M-N( — + d tan0)

cos0

Substituting, as before, for M and N, we have

V

MA=(B+R-WPsin0-Kuitan0)d-(WP+-i— -)(l-Zcos0)
y r cos0

Hence, tbe reaction of the upper recoiling
parts on the lower, after the powder period,
that is during the retardation of tbe upper recoil-
ing parts, is equivalent to:

(1) A force through the center of
gravity of the recoiling parts
parallel to the bore

B+R-Wrsia0-Ky i tan0

(2) A vertical force through the
center of gravity of the recoil-
ing parts:

.,. ,„,„' Br+ V_

cos0

Tbe mutual reaction between tbe upper and loner
recoiling parts, can be determined immediately as
follows:

(1) B+R along the bore

(2) N normal to the bore

(3) M a couple between the parts
Now, N=Wrcos0+Ky i (algebraically)

K;

fr-Wrsin0+ — Kyitan0 (vectorially)

CQSJ0

812

and through the center of gravity of the recoiling
parts. Further B and R may be resolved into a
vector

B+R parallel to B and R through the center of

gravity of the recoiling parts and a couple
Bdb+Rdr. Hence combining B+R and M, we have

B+R+Bdb+Rdr+M=B+R (through the center of gravity

of the recoiling parts, parallel

_ to the bore)

since M=-Bdb-Rdr

Combining, the parallel components through the
center of gravity of the recoiling parts, we have
B+R-WrsingJ-Kyitan0, along the bore

Kyi

Wr+ vertically

COS0

-(bdb+Rdr) a couple between the parts

Guide and clip reactions:

For the front clip reaction, we have

v»«-- 1

Bdb+Rdr
=(Wrcos0+Ky i)»t~( ) (Ibs) acting upward

on recoiling

parts, and

for the rear clip reaction, N *Nx *-

Bdb+Rdr

»(Wrcos0+Ky i)xt+( — ') (Ibs) acting up-
ward on recoil'
ing parts.

VARIABLE BRAKING ON UPPER Usually we have a
RECOIL BRAKE DOUBLE RSCOIL given upper recoil
SYSTEM. system with a constant

braking for the lower
recoil system, since

given single recoil mounts are converted into a
double recoil system by allowing the top carriage
to slide along an inclined plane. Further, in the

813

design of a double recoil system, since at high
elevations of the gun, the component of the re-
action of the upper recoiling parts along the
plane is small, the movement up the plane there-
fore, becomes relatively small. Hence in a design
layout the throttling grooves of the upper recoil
system nay be calculated on the basis of a single
recoil at maximum elevation.

We have therefore a very important class of
double recoil systems, where the upper recoil
throttling is based on a single or static recoil
and the lower recoil braking is designed for an
approximate constant resistance at minimum ele-
vation.

The recoil braking of the upper recoil system
consists of the sum of the following components:

(1) The recuperator reaction, which
is a function of the relative dis-
placement between the gun and top

carriage.

(2) The throttling reaction, which
is proportional to the square of
the relative velocity at a given

relative displacement.

(3) The guide and packing frictions,
which depend upon the normal reaction
between the parts, etc., but can be
assumed approximately constant.

The lower recoil braking will be assumed

constant at minimum elevation.

Considering fig. (9)
Let

Pb* powder reaction (Ibs)

B * total braking (Ibs)

Pf * total friction (assumed constant) (Ibs)

Ph « hydraulic braking (Ibs)

p » Pa+Ph+Pf

814

Png = hydraulic pull fron static force diagram

Pa= recuperator reaction (Ibs)

N^normal reaction between upper and lower recoil-
ing parts (Ibs)

R » brake reaction of lower recoil system (Ibs)
0= angle of elevation

9* angle of inclination of inclined plane
Wr= weight of upper recoiling parts (Ibs)
n*c= weight of lower recoiling parts (Ibs)
Vf= free velocity of recoil (ft/sec)
Vr»absolute velocity of recoiling parts parallel

to axis of gun (ft/sec)
Vc= velocity of lower recoiling parts along the

inclined plane (ft/sec)
Z = relative displacement between upper and lower

recoiling parts (ft)
Xc = displacement of top carriage up inclined plane

(ft)

Xr = absoluts displacement of gun parallel to axis
of bore (ft)

B'= counter recoil buffer rsactnon

During the powder pressure period:

On the upper recoiling parts, we
-p+Wrsin0=BrLJLL (l)along the bor

Pb

dt«

N-Wrcos0=mP (2) normal to the bore.

dta

On the lower recoiling parts, we have

daxc

Pcos(0+e)-Nsin(0+e)-R-Hcsine»nic (3) up the

dt8

inclined plans. We have further, the following
kinematical relations:-

dxr

«*^^«"«^E v + w/*rtof0? + fti Y~-Y A Y

V 1* — i ^^ Vy»o"lTV^COS\iL/Toy Aw»"™Ay»ol*A/»

'I ^j-^ tCi u r i c -L C

— - = Vcsin(0+e) : Yr=ycsin(0+6)

815

d*xp dvrel dv
hence - = - + — cos (0+6)
dt2 dt dt

d*yr dvc

- = - sin(0+6)
dta dt

Now between any two instants tn_j and tn we have,
from eq. (1)

tn Pbdt p-Hrsin0

— ~ (— ^ - )(tn-tn-l>'V? -V?-1
mr mr

.
p-W

r p-W_sinCf

therefore Vj-Vj'1 +(Vfn-Vf ^^-(-g- - )A tn

which gives a "point by point" method for determin
ing the absolute velocity of the gun parallel to
the bore.

Now if we substitute for the normal reaction
N in (3) its value

dvc
N *t»rcos0+inr —- sin(0+6)

u t

we have

P cos(0+9)-Wrsin(0+6)cos0-Br

dt

dt
hence

dvc p cos(0+6)-Wcsine-Wrsin(0+6)cos0-F
dt m +m_sin*(0+9)

w i

and between instants tn-i and tn, we have

mc+mrsin«(0*9)

A t

The total braking P, becomes P^P^Pw + Pf (Ibs)
In the static or single recoil, the top car-

816

riage stationary, we have Pns»Co ""* —

"xn«
NOII for the same relative displacement between

the upper and lower recoiling parts for the double
recoil, the throttling area is the same, namely

wxn, then _« v

vrel vrel*

Ph=co — bence Ph=Phs — —
"In *•

Therefore, from a static force diagram,
knowing the relative displacement = static recoil
displacement, we may determine Phs and v| . If
'1TS1 has been determined for the point, the
hydraulic braking is readily determined from
the above equation.

The recuperator reaction is determined from
the static force diagram when the relative dis-
placement is known. When the upper recoiling
parts begin to counter recoil relatively to ths
lower recoiling parts, we have

v« "Pf

Procedure for recoil calculations

We must first construct a static force and
velocity diagram for the upper recoil system as
would occur if the mount had a single recoil,
the top carriage being fixed. Let

v0 * nuzzle velocity (ft/sec.)

u* travel up bore (ft)

w = weight of projectile (Ibs)

w = weight of charge (Ibs)

Pa=max. total powder reaction (lbs)wv»

then average pressure on breech Pe * — ~~* (Ibs)

2gu

Pressure on breech when shot leaves muzzle —

27 u

Pob « — b« 1.18 PB (Ibs)

4 (b+u)»

.here b-(2Z £ - l)t /I- g £>• -1 (ft) ^ „

3

Time of travel to bore to 3 T~ uo (sec)

wv0+4700w
Max. free velocity of recoil Vf * - (ft/sec)

"r
Free velocity of recoil when shot leaves muzzle —

(w+0.5w)v0

(ft/sec)

Time during expansion of powder gases —
2(Vf-V0) wr

ti°= — r (sec)

Total powder period T = t. +t0 (sec)

*o

Free displacement of recoil during travel up bore —

w+0.5w

Free displacement during expansion of gases •—

Pob (T-tQ)«

«f -o - — « 3 +vfo<T-t0

r

Total displacement of free recoil during powder
period — B=xfo*xf'o

Three points are sufficient to establish approximate
the velocity curve during the powder period. They
•ay be taken at times to t\m and T respectively.

The total resistance to recoil for constant
resistance to recoil,

;•>*!

b-E+VfT

for variable resistance to recoil
mrV|+m(b-E)«

2[b-E+VfT- | — (b-E)]

818

At t0 when the shot leaves the nuzzle —

. Mo

Vo~Vfo~ mr

At t,jj when ire have max. restrained recoil
velocity, Koto

vnTvfn -- (ft/sec)
•r

K ta

where Vfm»Vf Q+Pob (tm-tQ)tl -- ] (ft/sec)

4fflr(Vf-V0)

K(T-t0)
tm « T (sec)

pob
At tine T, the end of the powder period — *

K T
Vr=Vf- — ( ft/sec)

mr

K0T2

Er=X£ (ft)

2mr

After the powder period, during the retardation,
we have for constant resistance to recoil,

V = /— £(b-X) (ft/sec)
mr

for variable resistance to recoil,

/(K - f(b+X-2Er)(b-X))

v = / 2_2 £ (f

mr

where b = the total length of static recoil (ft)

819

a = Cs — ; Cs=0.85 approx.; b= perpendicular

distance froa

spade to line of action of K. --*
Construction of static force diagram:

We have, for a constant resistance through-
out recoil, K*Pbs+Pa+Pf-Wrsin0 hence Phs*pa*pf=
K+Wrsin0 (a constant)
For variable recoil, in battery K SKO, out of

batteryK =k
where k = Ko-m(b-Er) and K = KQ during the powder

period .

=K0-m(x-Er)=k+m(b-x)
hence Pbs+Pa+Pf =K0+tfrsin0-fl>(X-Er)

r

r

Value of components Pf,Pa and P0. For a first
approximation, the friction component becomes,
Pf=0.2Wrcos0+p (estimated packing friction) and
will be assumed constant. The recuperator re—
action becomes, p _p .

Pa=Pai+ - X for springs
a <*± jj

where Pai=total initial spring reaction
Pa£=total final spring reaction

Vo k
Pa=pai ( - ) "here k=l.l to 1.3

v0-v

Vrt=initial volume (cu.ft)

rt
0

•

i i

8

Av = effective area of recuperator piston

(sq.in)

Pai= initial air pressure (Ibs/sq.in)
mo = ratio of compression (from 1.5 to 2)
The hydraulic throttling reaction, becomes for
constant recoil, Pns=(K+Wrsin0)-Pa-Pf

820

for variable recoil Pns=K0+Wrsin0-m(X-Er)-Pa-Pf

where the value of Pa corresponds to the displace-
ment X.
Construction of static counter recoil diagram:

The counter recoil may be divided into
and acceleration period, controlled or regulated
by a throttling resistance through a constant
orifice, and the retardation period where the

recoiling mass is brought to rest into battery

by a constant resistance to recoil, with a varying

buffer throttling. If

Pa=tbe recuperator reaction

Pf-total friction of counter recoil assumed
the same as for recoil and constant.

Bs=static buffer reaction

lo=length of constant orifice period (ft)

lb=length of variable orifice period (ft)
Then during the acceleration,

dv vs

pa-Pf-Wrsin0-B^=mrv — where B^= co — («0= a con-

U X «M£

0 stant)
and during the retardation

_ f* I TJ X

dv covs
B£ +Wrsin 0 + Pj -Pa = - mr v — where B^= — —

•i "*

Now — nay be determined by assuming a max.

wo

counter recoil velocity » 3.5 ft/sec.

at max. velocity, we have,

e i

Pa-Pf-Wrsin0- (-r)vjs = 0 and assuming v*s,
»5 c,

we readily determine — * G

*

The velocity and force curve during the first
period may therefore be constructed as follows:
(1) Plot the recuperator reaction
against counter recoil displace-
ment, that is,

821

V*,

V0-Av(b-X)

b = length of re-
coil

(2) Assume P£ =(0.2Wrcos0)

(estimated

packing friction) Con-
stant for the counter
recoil.

(3) Divide the acceleration period
into "n" intervals and take the mean
air pressure for this interval.
Then, knowing the velocity at the

beginning of the inta1"**!* *e can
compute the velocity at the end of
the interval by the formula, -

log (A- -— )slog(A- -j-^) p

where A = Pa-Pf-Wrsin0

co

— = G and determined as outline above.

(4) .Next construct from the velocity
curve a static buffer against
counter recoil displacement, that
is

co

Bs'(— ) v«
»o

The velocity and force curve during the re-
tardation period of counter recoil may be con-
structed, as follower-
CD The total resistance to counter
recoil being assumed constant
during this period, we have
Bs+Wrsin0+Pf-Pa=Kv whence, / 2Kv(b-x)

v = / —

822

A/

rig. 9

823

- m v*

T* m

where Ky = and vm is determined from

1^ the previous point by point

method to the end of the dis-
placement 10. Then, the velocity and buffer force
against recoil displacement is determined, since

Jrarvm vo k
B^ = J_£JL_ +p ( 2 ]K -Pf-Wrsin0

V0-Ay(b-X)

/ 2Kv(b-X)
and v« / where K

v

Dynamical equations of double recoil for point by
point method of procedure for construction of re-
action and velocity plots:

Let 0 = min. angle of elevation of gun

P = total pull between upward and lower re-
coiling parts (Ibs)
Phg- static hydraulic pull (Ibs)
Pa= recuperator reaction (Ibs)
F£= total friction assumed constant (Ibs)
Vj=free velocity of recoil (ft/sec)
Vr=velocity of upper recoiling parts parallel

to upper guides (ft/sec)
Vrej= relative velocity between upper and

lower recoiling parts (ft/sec)
V*o= velocity of lower recoiling parts up

plane (ft/sec)
X = displacement of top carriage up inclined

plane (ft)
B£ = static counter recoil buffer reaction

(Ibs)

R = lower recoil reaction parallel to in-
clined plane (Ibs;
Then, during the powder pressure period,

'+Pa+Pf (1)

824

P-W sin0
Vg =• YJT1 (V£ - Vg"1 ) • At (2)

mr

[Pcos(0+8)-Wcsin9-R-Wrcos0sin (0+8)j At

vn = ya-1 + (3)

mc+mrsina (0+9)

Vrel=Vr-Vccos(CJ+e) (4)

•- — — — ^- A t (5)

+ -£ A t (6)

c "c

2
After the powder period,

P-WPsin0
vn „ vn-i _ £ A t

After gun begins relative counter recoil,

p=p-

a- )_ P (

V8

In determining PQSvs and Pa the relative

displacement must be equal to the static dis-

placement of the recoil, that is *rel=xs» from
which we determine P v

825

240 M/M HOWITZER, GAS-ELECTRIC T7PB, DOUBLE RECOIL,

24° Elevation, R • 45,000 Ibs.

VELOCITY

DISPLACEMENT

P

I

T

T

6

c

R

R

u

0

n

0

0

u

a

e

e

p

i

t

t

t

n

r

1

1

n

e

a

a

r

A

a

t

r

1

1

P

i

t

t

p

a

V

•

a

I

i

1

*

T

B

r

g

V

V

a

1

i

r

a
1

e

e

e

n

"

e

a
k

1

7°

e

a
e
o

8

i

n
g

e

i

p

f
t.

0

o

i

o

F

a

n

o

6

n

d

r

u

e

•

c

n

e

8

1

i

d

a

During Powder Pressure Period

I

. OO4

151700

15.93

.703

15-332

.0307

.0014

2

.006

142400

32.849

1. 648

31.438

. 1710

.0084

3

.01

140000

41.431

3.310

38. 593

.5217

.0332

4

.012

138300

41. 351

5. 105

36.971

.9751

.0837

5

.016

128900

37.351

7.225

31. 161

1. 5201

. 1824

6

.02

117900

32. 801

9.405

25.051

2. 082

. 3487

7

.02

108000

28. 65

11.195

19.061

2.523

.5548

8

.02

96000

24.99

12. 51

14. 271

2. 856

.7919

9

.02

86500

21. 721

13.45

10. 191

3. 101

1.051

10

.02

77ioo

18.835

14. 03

6. 8O5

3.271

1. 326

11

.02

677oo

16.33

14. 23

4. 12

3.3804

1. 6091

12

.02

61000

14. 10

14.175

1.95

3. 441

1.893

13

.02

56600

12.052

13.941

.092

3. 461

2. 174

14

.002

54900

11. 854

13.911

.076

3. 46

2. 202

Gun beginning to C'Roooil

15

.01

46300

11.03

13. 601

.64

3.457

2.339

16

.01

45430

10.233

13. 263

l. 137

3.449

2.474

17

.01

42300

9. 501

12. 86

1. 519

3.436

2. 605

18

.01

38790

8.841

12. 397

1. 779

3.419

2.731

19

.01

35680

8. 243

11. 868

1.937

3. 40

2.852

20

.0 1

33550

7. 689

11. 297

1. 981

3.38

2. 968

21

. 01

33070

7. 144

10. 717

2.056

3.36

3.078

22

.01

31600

6.63

lo. 10 7

2.03

3.34

3. 182

826

24O M/M HOKITZER, GAS-ELECTRIC TYPS, DOUBLE RECOIL,

' o o r. t i nu

ed)

23

.01

31*740

6. 113

9.5

2.037

3.32

3. 28

24

.01

31500

5. 6oi

8. 889

2.0 19

3.3

3.372

25

.0 U

3168O

4. 982

8. 159

2.018

3.276

3.474

26

.014

31350

4. 269

7. 299

1.981

3. 248

3.582

27

.016

31590

3. 446

6.323

1.974

3. 216

3. 690

28

. 0 18

31360

2.530

5-215

1.94

3. 181

3.794

29

.02

31490

1. §O6

3.99

1.914

3. 142

3.886

30

.02

31520

. 481

2.765

1.889

3. 104

3.954

31

.01

31470

.031

2. 153

1.878

3.085

3.979

32

.02

31350

1.O5

1.063

1.962

3.047

4.011

33

.016

30000

1.82

. 036

1.85

3.017

4. O2

240 M/M H01ITZER, TRACTOR MOUNT, DOUBLE RECOIL.

)° Elevation, R - 80OOO

VELOCITY

DISPLACEMENT

P

I

T

T

G

C

R

U

0

i

o

o

u

*

e

P

i

t

t

t

n

r

1

n

e

a

a

r

a

t

r

1

1

i

t

P

s

V

a

i

1

a

r

B

g

V

a

1

i

r

e

e

n

•

g

•

e

e

k

6

8

i

P

e

1

n

P

t

c

e

i.

l

5

a

r

n

o

e

r

o

e

During Powder Pressure Period

1

.004

152000

15. 76o

.753

15.011

.O300

.0015

2

. O04

144600

28. 578

1. 425

27. 16O

. 1143

.0059

3

. 004

143900

35.505

2.087

33.428

.2355

.0129

4

. 004

14O30O

39.258

2.713

36.558

.3755

.0225

5

.004

137600

41. 235

3.3C6

37. 943

. 5245

.0345

6

. OO4

13460O

42.035

3.869

38. 185

.6763

.0439

7

.003

133500

41. 355

4. 984

36. 400

.9751

.0834

After Powder Pressure Period

8

.02

125800

36. 225

7.303

28.965

1. 6288

.2072

827

240 M/M HOWITZER, TRACTOR MOUNT, DOUBLE EECCIL.

( Cent inaed)

9

.02

110800

31. "705

8.803

22.945

2. 1479

.3683

10

.02

10050O

27.605

9.753

17.905

2.5565

• 5539

11

.04

91300

20.145

1O . 6 2 1

9.575

2. 1063

.9613

12

.02

74400

17.105

10.138

7.005

3. 2722

1. 1689

13

.02

69000

14.290

9.360

4. 980

3-3920

1.3640

14

.02

64600

11. 650

8.340

3.360

3.4750

1. 5410

15

.02

61500

9. 140

7.150

2. 040

5.529

1. 696

16

.02

58300

6.760

5.790

1.010

3.560

1.825

ft

.023

56700

4. 100

4. 120

.0

3.5W

1.939

Gun beginning to C'Eecoil

18

.004

477oo

3.710

s.^so

. 0

. 1592

1.955

19

.004

477oo

3. 320

3.340

. 0

.1592

1.969

20

.004

477oo

2.930

2.950

..O05

. 1592

1.982

21

.004

477oo

2. §40

2. 560

-.010

.1592

1.993

22

. 004

477oo

2. IgO

2. 170

-.010

. 1592

2.003

23

. 004

477oo

1.760

1.780

-.010

.1592

2.010

24

.004

477oo

1.370

1.390

-.010

. 1592

2.016

25

. 004

477oo

.980

1.00

.015

.1592

2. 021

26

. 004

477oo

.590

.610

.016

.1593

2.021

2-7

. 004

477oo

. 200

. 220

.019

.1592

2.026

OP
Plane
(in)

Kxt

Kyt

Kx

Vs

Fh<-£HH

<1

0

145000

45000

62COO

1180OO

2

124000

32000

46OOO

96000

4

112000

26000

38000

82000

6

10 3 5 CO

21000

30000

7000O

8

9500O

17OOO

24000

58000

10

89000

13000

18000

49000

12

82000

1OOOO

14000

400OO

14

76000

7200

11000

32000

16

7OOOO

5000

77oo

240OO

18

64200

2100

4 COO

13000

20

60000

1000

1000

130OO

22

55OOO

- 2000

- 1500

8OOO

24

§2000

- 3000

- 2500

5000

26

50000

- 6000

- 8500

2000

828

Gun beginning to Counter Recoil

"•.

rrr "'.

«,

•

28

41000

- 8000

- 11000

O

30
36

42
46

38000
2*7000
25000
24800

-10000
-14000
- 1 60 0 0
-16000

-14000
-21000
-22000

-22000

12600
12*700
12800

48

24700

-15000

-21800

13000

1

Acceleration up plane.
In battery 60° Elevation.

v4» .102 ve=.207

a - .102

.292

average

aa=.105
.086

v4= .345
a = .053

a * .085

.086
ace. » = 43 I/sec. «

.002

Out of battery 60° Elevation
v

vt»6.550

6.550 va=6.350

ace. =

.050
.133

a =.150

9

v4=6.150
.020av. = .133

13.3 I/sec. a

.010

(Reversed )

In battery 30° Elevation

v = .318 v = .475 v « .600 v = .720
a = .318 a!= .157 a~» .125 a = .120

average * .180 ace. =

180
.002

90 I/sec.*

Out of battery 30° Elevation

v1= 8.630 va= 8.323 va=7*970 V4*7.600
a = .307 a = .353 a,= .370

123

.343
average = .343 ace. = ' * 34. sec.*

.010
(Reversed )

829

OF

830

Out of battery 30° Elevation

(1) Recoiling parts along bore
i 50000-15790x.5»42100.1bs.

(2) Recoiling parts up plane acceleration
- 34. ft/sec. »

15780

x 34 » 16700 Ibs. normal comp. 16700", 91355=

32 2

15200. Ibs.

5231
Top carriage up plane x 34 3

32 .2

162x34=5510. Ibs.

5513
Cradle up plane x 34=171x34=

33.2
5820 Ibs.

Stability of 240 Caterpillar.

Moments taken at 0° Elevation, Howitzer out
of battery, about a point under of rear track
sprocket.

(1) Weight of recoiling parts 15790x59.
Weight of cradle 5231 116

Weight of top carriage 5513 80
Weight of bottom carriage 5250 45

Weight of tractor 55000 128

Inertia of recoiling parts 58000 93
Inertia of recoiling parts 6700 59

Inertia of cradle 10820 86

Inertia of top carriage 11580 71

931,610

601,565

441,040

236,250

7,040,000

5,394,000

395,300

930,520
822,180

1,703,785
Inertia forces 60° Elevation in battery

(1) Along bore » 159000-15790 cos 30°(Recoiling parts)
15000-13765-145325 Ibs.

(2) Up plane » acceleration = 43. ft/sec.*

831

15790

x 43=21070 Ibs. normal comp. = 21070*. 81355

32.2

* 19250 Ibs. 5231

(3) Top carriage up plane x 123. =162. x43. =6966. Ibs.

32.2

5513

(4) Cradle up plane x 12. 3=171x43. =7350 Ibs.

32.2

Out of battery 60° Elevation

(1) Along bore recoiling parts = 70000-13675»56325

(2) Up plane recoiling parts acceleration « 13.3 ft/sec.*

790

13.3 = 6517. Ibs. normal comp. = 6517*. 91355

5950. Ibs.

5231

(3) Top carriage up plane £H± * 13.3 = 162 xis.3

32.2

(4) Crldle^p'plane |fi| * 13. 3 =171 x 2274. Ibs.
In battery 30° Elevation.

(1) Recoiling parts along bore 147000-15790". 5=139100

Ibs.

(2) Recoiling parts up plane. Acceleration 90 ft/sec.

1 S790

x 90=44200 Ibs. Normal comp. 44200*. 91355

32.2

= 40300 Ibs.

(3) T.C. up plane |22I x90=162.x90=14600. Ibs.

J o . o

5513

(4) Cradle up plane x90=171x90=15400 Ibs.

32.2

(2) About center line rearmost roller. (llOin. from
trunnions )

Weight of recoiling parts 15790." 41. 647000.
Weight of cradle 5231. 97. 507000.
Weight of top carriage 5513. 62. 342000.
Weight of bottom carriage 5250. 27. 142000.
Weight of tractor 55000. 110.6050000.

832

Inertia of recoiling parts 58000. 93,

Inertia of recoiling parts 6700. 59,

Inertia of cradle 10820. 86,

Inertia of top carriage 11580. 71,

-5394000,

- 395000.

- 930000.

- 822000.

+ 147000

Direct Pads on Rollers.

In battery

Weight of recoiling parts 16700*. 99452
Weight of cradle 5231x. 99452

Weight of top carriage 5513*. 99452
Inertia of recoiling parts 17500*. 99452 +
Inertia of recoiling parts!40000x. 10453 +

Hydraulic resistance

58426,

f29967-28433+140000x .

1

17500x. 10452-26534. M045J

76200.

0° Out of battery.

Weight -

Inertia of recoil-
ing parts

Inertia of recoil-
ing parts

Hydraulic resistance

26534*. 99452 + 26389.
6700.x. 99452 - 6663.

58000x. 10453 + 6063.
25789.

J+22400 58000x. 99453 1
1+6700X. 10452-26534. *. 10452J

78330

60° In battery,

Weight 15790+5231+5513 .99452 + 26389.
Inertia of recoiling

parts (145324 x. 91355) + 132762.
Inertia of recoiling
parts (19250.x. 40674) - 7838.

166981

[6966+73 50+19250X. 91355
Hydraulic resistance <\

[I45325x.40674+26634x. 10453

24400 Ibs.

60° Out of battery

Weight (15790+5231+5513). 99452 + 26389
Inertia of recoiling parts(56325x

.91356)+ 51456
Inertia of .recoiling

parts (5950.x. 40674) 2420.

75425

&274+2155+56375X. 40674 "J

Hydraulic resistance > >-.

|5950x . 91355-26534x .10453]

32800

0° In. 80000 Out =

30° In. Out »

60° In. Out *

Weight of bottom carriage 5250 Ibs,

At 30° Elevation. Hydraulic resistance. In battery.
13 9000*. 80902-40300*. 58779-14600- 15400-26594

x. 10453
114000-23700-14600-15400-2780=57520

834

/L

/

112

,

z

\

835

836

837

"7

\

838

Bfi
If]

35

i

I

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-* — *r

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839

840

841

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Fig. IS

842

843

IfOO
1OOO

soo
o
soo

IOOO

/soo
xooo

24-00

3OOO

\

5 F

C*
VIM

»«-

TERPILLAR
HOWfT7ER

WO HIO» HIM

mr FRMMC

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00

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848

Hydraulic resistance out of battery

42100*. 80902+15200*. 58779+ 5510+5820-2780
34100* 8940 5510+ 5820-2780=51590. Ibs.

THEORY FOR VARIABLE RESISTANCE From the point
IN UPPER RECOIL AND CONSTANT •» by point method
RESISTANCE IN LOWER RECOIL as previously
SYSTEM. discussed in some

detail, we find,

that the resistance of the gun recoil system varies
from its static value in the battery position, to
very nearly the recuperator reaction plus the
total friction of the upper recoil system, the
throttling at the end of the upper recoil being
negligible and therefore the hydraulic braking
becoming zero in the upper recoil system; further
it was found that the gun recoil braking falls off
proportionally on the time. Let

PS = static braking for gun recoil system =
initial braking reaction on gun recoil

system (Ibs)
Pa£ * final or out of battery recuperator

reaction for upper recoil system (Ibs)
Pf * final braking of reaction on gun recoil

system (Ibs)
Rt - total friction of upper recoil system

(Ibs)

lfr = weight of recoiling parts (Ibs)
HC = weight of top carriage (Ibs)
V = initial upper recoil velocity (ft/sec)
Z * displacement of gun on carriage (ft)
N * upper normal reaction between top car-
riage and inclined plane (Ibs)
R * lower recoil resistance parallel to in-
clined plane (Ibs)

X = total run upon inclined plane (ft)
v = velocity of combined recoil Rel. vsls.=0

t1 * prime P0 common recoil

349

The mean braking zone for the upper recoiling

parts, becomes, p .p .p
rs af t

2

Further the distance run up the inclined plane
during the time tj was found to be approximately
X = - vitl. The approximate equations for the

double recoil, with a variable re-
sistance in the upper recoil system and a constant
resistance for the lower, become,

ff V + to 4700
V=0.g( ) (1)

"r

W_ V-v cos (0+9)
P0-ffrsin0= — t ; 3 (2)

r viSn
N-Wrcos(? = —5

t

(3)

c t

P.cos(0+6)-Nsin(0+9)-w.sin 9-R= -- (4)

g t1

wr+wc
X = 1 vtf + - - v* (5)

2Rg

Z = - t1
2

DBRIYATIOK OF THB PYNAMICAI. EODATIOHS
POINT BY POIMT METHOD COMP UT AT I OH :

Total pull between upper and lower re-
coiling parts:

This reaction is composed of:- v

(1) the hydraulic braking pull =Pns( )*

vs

(Ibs)

(2) the recuperator reaction at the
relative displacement under con-
sideration — Pa (Ibs)

(3) the friction between the recoil-
ing parts — Pf (Ibs)

850

v«el

Hence P»Phs - +pa*pf (Ibs)
vs

REACTIONS ON THE UPPER RECOILING PARTS;

If Pb» the powder reaction, then for the gun
along its axis, we have,

Pb-mr - - P+Wrsin0=0 (1)

d v

and normal to its axis

d»y'
N-mr -— -- Wrcos0 =0 (2)

Integrating equation (1), we have,

tm Pbdt P-W sin0

— - ( - - - ) A t= A vr
l- m

P-W sin0

vn _ vn-i _( - i - )A t, vn . vn
f f m r r

bence P-Wrsin0)

vn = vn-i +(yn_vn-i)_( - £ -

r r f f m

From a somewhat different point of view, we have
from (1)

r, , «, «
HP*W.«ini»0 since

d * d x

~ni..

dv

' /%

+ — — cos(0+9) See acceleration diagram,
dt

d*xrel dvc

then Pb-nPt + cos (0+e)]-P + wrsin0+e

dta dt

Integrating, we have

pbdt P-Wrsin0

m -( TO t!*vrel*vc cos(0+9)=vr hence, as

before,

yn _ yn~i+ (yn_yn— 1 \ _

mr
Fro« the vector disgram of acceleration,

851

dav' dv

= sin(0+9) hence equation (2) becomes,

dt8 dt

dvc
N-mr sin (0+6)-W Cos0 = 0 (2a)

dt

REACTIONS ON THS LOWER RECOILING PARTS

These reactions are N and P reversed (the
mutual couple having no effect on the translation)
of the upper recoiling parts, the braking reaction
K of the lower recoil brake and the weight and
kinetic reactions of the top carriage.

The normal reaction and couple exerted by
the plane has no effect on the motion of the sys-
tem, then, along the inclined plane,

Pcos(0+9)-Nsin(0+9)-Wcsin e-R-no ~jf~ = C

dvc
Substituting N =mr — ; — sin(0+9 )+Wrcos0

01 T

we have ^v

Pcos(0+9)-m --4 sin2 («f+6)-W_sin 6-tfrcos0sin(0+9)
d t c

-R-tDc •• = 0, combining terms and simplify—
dt

ing, we have

ii£ -p

r c dt c ' i

hence

dvc Pcos(0+9)-Wcsin9-Wrsin(0+9)COs0-R

dt mrsin2 (0+9)+aic

and between any two intervals,

Pcos(0+6)-W sin9-W_cos0sin(0+6)-R

•j.D-Ttn-l, r . . 1.

V

GEOMETRICAL RELATIONS,

To compute P it is necessary to compute the
relative velocity and displacement respectively
for any given interval in the recoil. Obviously
from a velocity diagram

852

vrel=vr-vcco3(0+9) and the relative displacement
vrel~xref + ? A l and the displacement

C C

up the inclined plane x£ = xj}"1 + ---•' - A t

METHOD OF COMPUTATIOH,

Knowing v°-1 , vp"1 and vjjgj at the beginning
of the interval, we have,
vn-i

1*6 1

P=Phs ^ — ^"^ +pa*pf at relative displacement xg^J
v«.

then v^^v"'1 +[ - . . ,^ QX - 3A t
c c m+msin2 (0+9)

and _

vnsyn-l -

r

""

From these values, we have v?el = vr"~vc
and therefore

vn +vn-l
tfn _ xn-i vrel+vrel

xrel - rel + - - - A l
o

and 1

After the powder period, obviously the expression
for vr reduces to,

P-Wrsin0
m

vnsyn-i _ ( - L - )A t

HELATIVB COUNTER RECOIL OP THE UPPER

RECOILING PARTS:

In the expression for P, the hydraulic re-
action and friction reverses. If B1 is the c're-

coil buffer force in the upper recoil system at
a given relative displacement, then

v a. S-..-T?

853

Pn= - B'("l — ) i pf=~pf assuming friction the same.
vs

hence, „« ,
y rel
P=Pa-B'( - — )~Pf (Ibs) The remaining expressions

are the same as before.

This method of computation is sufficiently
accurate and was followed in the recoil cal-
culations illustrated.

APPROXIMATE CALCULATIOMS FOB STABILITT
WITH A DOUBLE RECOIL.

Reactions and velocity for double recoil
system:

P = resistance of gun recoil system

Wr* weight of recoiling parts (upper)

Wc= weight of top carriage and cradle (lower)

V = initial velocity

z = displacement of gun on carriage

R * reaction of lower recoil system

N = upper normal reaction between recoiling parts
and top carriage

M « lower normal reaction between top carriage
and tractor.

X = total run up on inclined plane.

v = velocity of combined recoil

t = corresponding tine.

0 = angle of elevation of gun

9 s inclination of plane.

Values assumed for computation of recoil:
P *
R =

9 = 6°

Wr= 15,790 Ibs.
«c=ll,570 Ibs.
0 =
General equations for double recoil:

Wr V-v cos (0+9)

P-Hrsin0= — [ - - - ] (1)
g

854

wr v sin (9+0)
N-Wrcos0» (2)

g w

Pcos(0+9)-Nsin(0+9)-Wcsin9-R» — - (3)

[Ncos (0+9 )+Hccos9+Psin (0+9 )-M- 0] (4)

v *r*we
B 2 ** 2Rg **

Pz+ 7(-^j^) v*+R | t » j mrVa (6)

wxv_+4700w
V-0.9C ) (7)

»= | t (8)

Energy equation:

PX0= ^mr[vk-?acosa(0+e)] Indication of P

Nxsin(0+9)= ~ mrva(0+9)
t r

[Pcos(0+9)-Nsin(0+9)-R]x« jMcv2
0(X_-Xcos(0+9)]+ i mrvasin2 (0+9)+R x = -rar

t» z r I i

[V«-?2cos2(0+9)]- i m.v2

2 l»

a t Rvt i
hence Pz+ -mrva+ ^v2* — mrV2

Further wxvm+4700* w

V » 0.9( ) where w = weight

Hr of shot,

w = weight of

powder

Wr= weight of recoiling parts
vm * muzzle velocity of shot

240 M/M DOUBLB RECOIL MOUNTED OX UARK III MI
CATERPILLAR.

SCKMZIDER HOWITZER AT 0* ELEVATIOH OF HOWITZER.

Given: Wr * 15780 Ibs. — weight of recoiling parts
Hc = 11570 Ibs. — weight of sliding carriage

855

9=6° — angle of inclined plane.

V - 45 ft/ffi — max. velocity of upper recoil-
ing parts at beginning of re-
coil.
R * 80000 Its. — resistance to recoil on lower

recoil system.
From static force,

Diagram 240 M/W Howitzer,
Ps = 155000 max. pull

Rt+Paf =60,000 maximum recuperator reaction plus
friction at end of recoil.

Approximate Calculations,

Ps+Paf+Rt
P0 » - - - whence Ps=155,000 (Ibs)

Paf+Rt = 60,000 (Ibs)

|215,000
hence Pc = 107,500 Ibs. mean reaction

= 1.480+0.58= 2.06 ft.
Z * 22.5x0.158= 3.56 ft. = 42.7 in.
Check on Z by energy method:

t, 27360,-- — » 80,000*10.4
107,500 Z = 2("327F) = ~~^i - X

- I 1579° Tf
~~- ^ 32.2

107, 500Z+46, 000+68, 200=487, 000
Z-3.,56 ft. Cheek

R = 80,000 Ibs. For horizontal recoil, -0= 0
e=6° sin 9 « 0.1045 cos 9 = 0.9945
45-0.9945,

107.500

32.2

856

1S790 v

N-15790 ~~ * 0.1045 - (2)

11570 v

0,9845x107, 500-NxO. 1045-11570x0. 1045-80, 000 =—

32 .2 t

107, 000-0. 1045N-1210-80, 000=359 -

v 25, 790-0. 1045N N-15,790 L

-- ' » — -£ hence 7.11N=136,290

t ooV DJ. . ft

H » 19,170 Ibs
51.2 v 490 (45-0. 9945v ) 51.2 v

N-!5790 107,500

51.2v-692-15.3v

66.5v=692 hence v - 10.4 ft/sec.

51. 2x10. f

3380

-1575

Total time = T+t*. 032. 158=. 190 sec.

240 M/M POUBH RKCOIL MOOHTgD OH MARK IV MI

CATKRPILLAB.

"~~
APPBOIIMAT1 CALCPLATIOHS TOR 240 M/M QAS-KLECTBIC

DOOBLB HICOIL 8TSTKM AT 24° 1LKVATIOK OF HOWITZIB.

Given. Ir » 15790 Ibs. — weight of recoiling parts.
Hc * 11570 Ibs. — weight of sliding carriage.
9 » 7e — angle of inclined plane.
V » 45 ft/a. — max. velocity of upper re-
coiling parts at beginning

of recoil.
R = 45,000 Ibs. — resistance to recoil on

lower recoil system.
From static force,

Diagram 240 M/K Howitzer
Pg = 155,000 max. pull
Rt +Paf * 60,000 maximum recuperator reaction

plus friction at end of recoil.
Approximate calculations,

P0 » -= — — whence Pa « 155,000 Ibs.

857

P R - 6°'000

"aft" ~

[215,000

107,500 Ibs.

R = 45,000 Ibs., 0 * 24° Elevation of gun, 6= 7° —
angle of inclined plane
0+6= 31° sin(0+9)=.5150 cos (0+6)= . 3572
sin 9 = .1219 Wrsin0=1578Qx. 515=6140 Ibs.

107, 500=6140=490 (40"°'85y2V) (1)

t

N-15,790x.8572=4SOx >515° V (2)

107, 500x.8572-N. 515-11, 570x. 1219-45, 000-359 - (3)

v

359 - = 93, 200-0. 515N-1410-45, 000

v _ t46, 800-0. 515 N _ N-13520
t " 359 252

46,800-0.515 N=1.425N-19250
K = 34,000 Ibs.

252 v 490(45-0.8572 v)

N-13,520 107,500-6140

252 v 22080-420v

SM80 101,360

337v» 4480 hence v = 13.23 ft/sec.

252x13.23

* * nn *c,n ~ 0.163 sec.
20,480

Total time T+t»0.163+. 032=0. 195 sec,

s_ 27360 13.23

x » -(13.23xQ.195) + - -
4 2x32.2 45000

x = 1.955+1.65=3.59 ft. » 43 (in)

v 45

Z = - t = — xQ.163- 3.67 ft. =44 in. check.

6 Z

107,500 Z * J(£Z££2) I3T232 * ^221 x 73.23 x 0.163
32.2 2

32.2
74, 500+48, 500=497, 000

858

Z » 3.48 ft.

The discrepancy between this value of Z of
the above is due to the fact that work done by
gravity is omitted in the energy equations.

Theory of stability not braked.

By D'Alerobert 's principle, we have

K * P —IB
° ' r dt«

(1)
(2)

where KQ * dynamic inertia resistance of recoil-
ing parts * 0.9 K (assembled approximate)
F = tractive force reaction
r * radius of traction rim

859

d29 d29 d29

(F-BF)+r-Rr0»I0 TTT * J0 TE* +»«! 8(7-78)
w v at* dt2

(2)
(3)

r'

(4)

(5)

It=2nr mr2

Hence we have the following equations:

(1)

•v

d2x
(F-2F1) r-Rr0=[I0-I0+3ar2(itr+l)}— (2)

r

-^ ** (3>

^ A * fl

(4)

f c\

i- 'r2*At W

The reaction of the truck rollers on track,

= 40.4^2-

i dt«

4

The reaction of the clutch shaft pinion,

,i. 1.75 24 20.35 dax dax dax

R (-)* -- - - = 19.1 - hence R » -

»V 1.43 5.2 6.00 dt» dt2 » dt»

The reaction of the drive gear pinion,

. 7.32

860

d*x d*x

/. R(0. 2165 )-(76. 4*0. 846)-— =7.32 — 4

at* dt

d*x
R (0.2165 )-(64. 6+7. 32)—-

R - — • = 332.0

0.2165 dt*

R = 331.0 — - 331 .-^
dta dt*

d*x

and I+2mr*Ur+l)

I0»35.54 m= ~ = 4.65

1^=18.5 r = 1.43 ft. 1 = 158 in. or 13.2 ft,

bence B

35.54*9.30x1.43 (nl. 43*13. 2) d^x 18.5 dax

1.43 dta 1.25 dt*

(p-40.4 Hi)i.43-331 £i = 274.8 Hi

dta dt*

663.4 d*x , d*x

bence p = _._ _ = 463 —

2l!259 + 15000 _ ^ m 838

32.2 32.2

d2x d2x
50000-463 T-T=838—

dax 50000

- » - = 38.4 ft/sec.*

dt* 1301

dt * * dt*

— )V*
r

R-(ir*i0k-*irkj ~)(v

dx

861

Check on equivalent mass of rotating parts:

Kinetic energy of rotating parts in terms of
translatory mass,

+7 lO.l(^j)* + J- 18-5(^j)*+7 9.3(nl.43+13.2 v»

t 28*48. 6+430

J 10.1 Sj — =20.2

t 18.5

• \t

82.2

231.21
I Mrott

Mrot = 463.0

CALCOLATIOH OF* STABILITY.

Evaluation of inertia couples:

d2x

Track rollers 20.2 = 653

dta

dax

Track inertia sprockets 274.8 — - = 8900

d t

Intermediate gear 7.32 =237

dt«

Clutch 19.1 —=616
dt«

Resultant couple effect;

653 10,169

8900 237

616 9,932 ft.lbs. stabilizing moment, due to

119,000 ft.lbs inertia couples of

10169 wheels:

862

Kd=0.9K=500001bs. Overturning moment.

50000x72=3, 600, 000 ( overturning moment)lbs.

Stabilizing moment: 6248x69.5=2,480,000 3,520,000
9396x111* 1,040,000 119,000

3,520,000 3,639,000
Mc — - = x 32.3 = 36400.

36400x32.5=1,185,000

3,639,000
1,185,000

4,824,000
Dynamic

Overturning moment 3,600,000 Ibs.
Stabilizing moment 4,824,000 Ibs.

Static

Overturning moment 4,100,000
Stabilizing moment 3,520,000

d*x

50000-15000-4647-*-= 1270

d t d t

d*x d8x

3 500 0*1 73 4r— T hence -T-T » 20.2 ft/sec.
at dt'

292x38.61
Ft = M V t = - = 0.226 sec.

50000
, «0.«'.8»6 . 2.28 ft.

S = S^ St = 2.28
.87

3.15

CHAPTER XIII
MISCELLANEOUS PROBLEMS AND TYPES OF CARRIAGES.

GENERAL DYNAMIC EQUATION 01 RECOIL The follon-
DURING POWDEF PRESSURE PERIOD. ing theory is

perfectly
general and
specially ap-
plicable for types of mounts that do not recoil
along the axis of the bore. Let
m - mass of projectile
i =» mass of the powder charge
v = absolute velocity of the shot up the bore

ft/sec.
vx» component of v parallel to recoil path

ft/sec.
vy= component of v normal to recoil path

ft/sec.
vrel * relative velocity of the shot in the

bore

mr= mass of the recoiling parts
P = mean powder force
0 = angle between axis of bore and path of

recoil
Ng= normal reaction between projectile

R = total resistance of the recoil system (Ibs)
u = travel up the bore (ft)
X = retarded recoil displacement (ft)
Xf = free recoil displacement (ft)
B = angle between absolute velocity of pro-
jectile v and path of recoil.

Assume half the charge to move with the projectile

and half with the gun.

The reaction between the gun and projectile,

becomes P cos 0-Ngsin0 along the bore

863

864

P sin0-Ngcos0 normal to the bore.

The equation of motion of the recoiling parts,
becomes, along the recoil path,

dV
Pcos0-Nssin0-R= (mr + 0.5 ID ) — Integrating and

Pcos0-Nssin0 Rdt dividing by mr,

we have ( n c- )dt- . __ = V Now from
m_+0.5m mr+0.5m

the vector

diagram of velocities, we have, adding vectorily,

vrej+V»v but since V = Vj approx. that is the

retarded velocity of recoil is approx-
imately equal to the free velocity of recoil, we
have vrQi + Vf= v (approx.). Now in the free re—

coil Pcos0-Nssin0

( - _ - )dt=V# that is the expression
»r+°-5 m Pcos0-Nsin0

is

v mp+0.5m

measured

by Vf and which assumes, for given intervals of
time P and Ns are not greatly different in the
free recoil as compared with the retarded recoil.
If R was sufficiently great to prevent an appreciable
recoil Ng would disappear but P would not vary even
then greatly for given intervals of tine between
free and stationary recoil. Further Ns is small
even in free recoil as compared with P, hence the
above expression would be but slightly modified.

Next, considering the motion of the projectile
in a direction parallel to the recoil, we have

(Pcos0-N8sin0)dt=(m+0.5)vx but since *xa»relcos0-Vf
we have (Pcos0-Nssin0)dt=(m+0.5m) (vpelcos0-Vf )

Combining with the expression for free recoil of

the recoiling parts, (mr+0.5l)Vf3(m+0.5m)(vrejcos0-Vf )

Hence, V* =

cos0

Since 8 equals the angle v makes with the recoil

865

path, we also have (Pcos0-Nssin0)dt=(m+0.5i)vcosB

and therefore (mr+0.5m)Vf>(m+0.5i)vcosB

ffl + 0 . Sm p

hence Vf» m +Q.5I Now B differs very little

from 0, and assuming 8=0

hardly modifies the recoil effect; further 0.5 m
is negligible as compared with mr. Hence

in +0.5 IB
Vj 3 - v cos 0 approx. The dynamic equations

of recoil, become

therefore

Pcos0-Nasin0

m+0.5m mr+0.5i
(m+0,5i)vreicos0 Et

mr+0.5i

m+0.5l Rt

« v cos0 (approx.)

mr mr

Integrating again X = fv*dt - - — = *_ cos0-

2mr mr+m+m

During the after effect period of the powder
gases, the reaction of the powder is approximately
along the axis of the bore and the procedure of
computation has been previously discussed in detail.

The effect of the reaction Ns is to deviate
the motion of the projectile, causing the projectile
to leave the muzzle at an angle somewhat greater
than the angle 0.

To compute this angle, we have, v sin B-vre^sin0

vrel /Br+a*gv/m+0.5i ,cosB
hence sin B = — — sin0=( )( )

v m+0.5m nr+0.5i cos0

mr+m+i «j mr+m+I

tan B= — — — tan0 and B * tan (————) tan0

mr+0.5l mr+0.5i

The increase in ths apparent angle of elevation
becomes B-0 and is usually small and may be neglected
in recoil problems.

On the other hand, to compute Ns is important
since it causes an additional load on the elevat-
ing mechanism during the travel of the shot up the bore

866

HS-

dV ^

(m+O.Sro)2

dv

(•+0.5i)*

mr
* dv

dt

2»r
• •»0.5l

dt

o-i n 90(P -i n

- vm+u. om ;••

- (approx)
t

•fKA

normal reaction of

the projectile when a gun recoils at an angle (6
with the axis of the bore, is always proportional
to the powder reaction which varies from point to
point along the bore. Though the max. reaction
occurs practically at t~he beginning of recoil,
the moment is usually found greatest when the shot
reaches the muzzle of the gun.

REACTIONS AND GENERAL EQUATIONS Consider the
IN A FECOILING MOUNT. recoiling parts

to be constrained
in movement always
parallel to the
axis of the bore, the constraints being offered

by suitable guides or a gun sleeve fixed to the
cradle. We will assume rotation possible about

the axis of the trunnions. Let

Pfrs tbe powder reaction on the breech (Its)
Q^ and Q = the front and rear clip reactions

(Ibs)

tan M =t- the coefficient of guide friction
Mr and Wr - mass and weight of the recoil-
ing parts (Ibs)
B = total braking force (Ibs)
X and Y = the coraponents of the trunnion re-
action parallel and normal to tbe
axis of tbe bore (Ibs)
£ = elevating gear reaction (Ibs)
.) * distance from trunnion to line of action

of E (ft)

Mc and tfe = mass and weight of the cradle (Ibs)
6e - angle between E and axis of bore
vr * relative velocity of recoiling parts in
cradle (ft/sec)

867

»= angular velocity of tipping parts about the
trunnion (rad/sec)

Ir= moment of inertia of recoiling parts about
the center of gravity of the recoiling
parts.
Itr= moment of inertia of recoiling parts

about trunnion axis
Itc= moment of inertia of cradle about the

trunnions.

XQ and yQ = battery coordinates of the center
of gravity of the recoiling parts
with respect to the trunnion.

xc and yc = coordinates of the center of gravity
of the cradle with respect to the
trunnion.
d^b= distance from trunnion to line of action

of B.

T * /7[a+Y2 - total trunnion reaction,
r' = radius of trunnion bearing
nt * friction angle in the trunnion bearing
x x y and y - coordinates of the front and
rear clip reactions with
respect to the trunnions.

R8ACTIOHS OH THE RECOILIMG PARTS.

The reactions on the recoiling parts, consist
of the reactions of the cradle QtQa and B, the
reaction of the ponder Pb and the various inertia
forces as shown in the diagram.

Referring to fig.(l) and considering the
motion of the recoiling parts assuming by D'Alemberts
principle, kinetic equilibrium, we have

(1) Along the axis of the gun

dw dvr

Pb-B-(Qt+Qt)sin u+l»rsin0-mrwa (xo-x)-mry0 mr— =0

d t

(2) Normal to the bore

(Qf-Qt)cos u-Hrcos0+mrway0-Br(x0-x) — +2mrwvr*0

dt

868

Fig. 1

869

(3) Moments about the trunnion,

dv

r dw

P(e+s>Bdtb-mr— s-I t —+2mrBvr(x0-x)-Wrcos0(x0-x)

u .xi-

where It=Ir+oirr» = Ir+mr[ (xo-x)* + y« and thus a
variable with the recoil.

REACTIONS OM THE CRADLE.

Referring to the cradle, we have the reactions
Q± Q8 and £ reversed, of the recoiling parts on the

cradle, the trunnion reaction divided into components
X and Y, the elevating gear reaction E and the

various inertia forces as shown in fig.(l)
passing through the center of gravity of the
cradle, together with the inertia cou pie Io — .
Referring to fig. (1) we have,

(I1) Along the direction of the bore
or guides

B + (Qt+Qt)sin u +Wcsin0+mcwfxc •'nigy,, — +Ecos 6e-X=0

dt

(21) Normal to the guides,
-(Q -Q )cosu-H_cos0+m,,xr -- m,,vr2y-+Esin6 + Y=0

21 t» **cj*. B - **

at

(3 ' ) For moments about the trunnion
axis,

dw

-Wesin0.yc-Bdtb-Ej-Itc -

where Itc=Ic + n!crc=

EXTERNAL REACTIOHS ON THE TIPPING PARTS

Assuming the tipping parts to be balanced
about the trunnions, which is customary in order
that the tipping parts may be rapidly elevated,

ve have Wrxc-Wcxc=0 mrx0-mcxc=0

and

and for the total weight of the tipping parts
Wt=Wr+Wc and Mt=mr+mc If now, we combine,

870

(1) and (I1), (2) and <2'),(3) and (3 ' ) and noting
the above relations, we will have for the kinetic
equilibrium of the tipping parts,

(1") along the bore

dvr
Pb+Wtsin0+mrw*x-mp--— +Ecos 9e-X=0

(2") normal to the bore
-Wtcos0-mrx — + 2mrwvr+Esin6Q+Y=0

Q. t-

(3") moments about the trunnion
Jvr dw d"

Pb(e + s)-mr -rr- s-It— • -ItcTT +2mrwvr(x0-x)+Wrx cos 0
dt

-Ej-Tr'sinut»0

Therefore, we have for the retardation, exerted
by the top carriage on the tipping parts,

For the trunnion reactions,

r

d t

Y»Wtcos0-Esin 9e-mr(x— +2w vr)

For the elevating gear reaction,

dw ^vr

Pb(e+s)-(It+Itc)— ' +Wrxcos^-Trlsinu1+mr[2wvr(xc-x)— -.s]

E=

j

APPLICATIONS OP THE PRECEEDING When the brake
FORMULAE. cylinders recoil

with the gun as
in the slide or
sleigh containing

the recoil cylinders and rigidly attached to the
gun used with the Schneider naterial, the center
of gravity of the recoiling parts falls considerably
below the axis of the bore. To offset the effect
of the large powder pressure couple and reduce the
reaction on the elevating arc, we may employ a
counterweight at the top of the gun to raise the

871

center of gravity nearer the axis of the bore as
was done on the 155 m/m Schneider Howitzer or we
may introduce a friction disk at the elevating
pinion, this allowing rotation of the tipping
parts about the trunnion.

In other types of mounts, a spring buffer
may be introduced in the elevating gear thus re-
ducing the elevating gear to a small finite value,
and the moment effect of the powder pressure couple
being distributed over a longer period.

If now we neglect, w*x and 2w vr as small and
during the powder pressure period x being small we
may neglect also, x -— and ffrx cos 0. The re-

actions on the tipping parts, become

dvr
X-Pb-ar— +Wtsin0+G cos 6e

Y*Wtcos0-Esin &6

dw d*r
and PfcCe+Bj-Tr'sinu^Ut+I^) - mr^~ 8

Now Pb

dw

where K= the total resistance to recoil during the
recoil neglecting the rotation effect during the
powder period.

If y0 is small, that is if the trunnions are
approximately on the axis of the bore, we have,

Pb=*r — JL = R (approx.)
dt

Assuming the brake disk on the elevating
pinion shaft to offer a given torque, we may
readily compute E. In other words, the pinion
bearing is designed for a given reaction. This
reaction should be comparable with the reaction
required in the out of battery position of the
recoiling parts.

872

Ks+Wrb cos0

That is E = c( ) where c = 2 to 3 de-
pending upon max.
allowable angular dis-
placement of tipping parts, where b = length of
recoil, K * resistance to recoil, s = distance
from K to trunnions. The trunnion reactions become

simply, X »K+Wtsin0+E cos 9 Y » W. cos 0-E 6
• t e

Thus the trunnion reactions are fairly in-
dependent of the rotation about the trunnions,
being primarily dependent only on the elevating
gear reaction, the total resistance to recoil and
the weight of the tipping parts. The additional

forces induced by rotation about the trunnion can
be treated as secondary forces. ~~~~~

The total trunnion reaction becomes, T=/V+Y*

(Ibs)

To determine the angular acceleration with a
given elevating gear reaction E. We have, approx.

Pbe+Ks-Tr'ain\it-Ej'(It + Itc) — hence

dt

Assuming Tr1 sin ut

1 Et

I

•n

Jt+Itc Ks and Ej as constant,
Ir'sin n +E

since obviously Tr'sinut+Ej must be greater than Ks.

Further since t
f

Pbdt «(m+0.5 n)v

where m = mass of projectile

m = mass of powder charge

v = velocity of projectile in bore (ft/sec)
we have

(m+0.5i)ve /Tr 'sinut+Ej-gs ^

w 3 ~—^— — — ~ (. ) t

873

e (m+0.5i)ve Tr ^

where u = travel up the bore, (ft) To allow for
the reaction effect of the powder gases, we will
assume the free angular displacement at the end
of the powder period, given by

(ui+2m) ve
9j = " Hence the angular velocity and

* tc displacement at the end of the
powder period become

(mv + 4700i) ,Tr 'sina +Ej-Ks.
- e - (_ - 1 - )t

e = (m*2m)ue _ Tr'simyEj-Ka t;

where t = total powder period (sec)

v = muzzle velocity of projectile (ft/sec)
u = travel up the bore (ft)

The remaining angular displacement is that due
to a constant torque (Tr'sinu1+Ej-Ks) acting on a
rotating mass with an initial angular velocity »t.
Hence (Tr 'sin u^Ej-Ks) (6t- 9t)= ^Ut+Itc) "»

and therefore, for the total angular displacement

6t

(m+2i)ue Tr 'sin^+Ej-Ks) t* (It + Itc^wi
tXIt + Itc it^tc 2 ^ 2(Tr'sinct+Ej-Ks)

mv+4700i) Tr'sinu^Ej-Ks
where w^ =( e -( ) t.

and t is computed by the methods of interior

ballistics and T= /x*+Y* using a suitable
value of E, we may compute from the above ex-
pressions the total angular twist.

GENERAL EQUATIONS:- ROTATION With rotation of

OF THE TRUNNIONS ABOUT A the trunnions about

FIXED AXIS OR A TRANSLATION an axis, the ele-

OF THE TRUNNIONS. vating gear reaction

is usually reversed
and the magnitude of the reversed action on the

874

**&

Fig.

876

elevating gear depends upon the product of the
angular acceleration about this axis and the total
moment of inertia about the trunnions of the tipping
parts. Thus, in the rolling of a ship or in the
jump of a field carriage where the angular ac-
celeration upon the mount may be considerable, and
with heavy tipping parts, a large reversed reaction
is exerted on the elevating gear, which in turn
modified the trunnion reactions. This same
phenomena occurs in a double recoil system, or in
a railway mount where the mount below the recoiling
parts is accelerated up in an inclined plane or
along the rails.

Let us first consider, the angular motion
induced in the tipping parts when the elevating
gear reaction is nil.

Assuming the trunnions to rotate about an axis
0, fig. (2) and the axis of the bore and center of
gravity of the recoiling parts to be along the
trunnion axis, then,

The Kinetic reactions on the tipping parts,

become

(1) The trunnion reaction X and Y
which impress the angular acceleration
on the tipping parts. Due to the
friction of the trunnions T = /X2+Y*
has a moment about the trunnion axis:

T r1 sin u.

(2) The tangential component of the
ine,rtia force of the tipping parts

= — R — — and its moment about the

g dt2 trunnion, axis becomes,

Wt

T R

W_x r

Let Tm= - sin(0+e) "hence — R1 - (Tm)= — R1
W g dt8 g

dt

876

(3) The centrifugal component of the

inertia force of the tipping parts *

wt dQ

— R (— — )• and its monent about the

g dt

trunnion axis becomes

dt g dt

(4) The rotational inertia couple of

tbe tipping parts
j _.

where w = tbe angular
velocity about the

trunnions, Itr= Inonient of inertia of

the recoiling parts about tbe

trunnions

Itc= moment of inertia of the cradle

about the trunnions
(5) The weight of tbe tipping parts,

its moment being Wt (TG)cos0=Wt
W_x

0

(- — )cos0=Wrx cos
wt

(6) Tbe complementary centrifugal
inertia force due to tbe relative
motion,of the recoiling parts38

2mr * where x * tbe relative

d t

displacement of tbe recoiling
parts. Its moment about tbe
trunnion becomes,

dx

2mr(x0-x) * —

(7) Tbe powder reaction, and tbe
relative inertia resistance due
to tbe relative accelerstion of
tbe recoiling parts. We are

not concerned with these reactions,
since their moment effect is nil,
it being assumed that their line

877

of action passes through the trun-
nion.

We have, therefore for the moment equation
about the trunnion, considering the kinetic
equilibrium of the various inertia forces,

Tr'sin a +(Itr+Itc)— - 2mrw— - -WrXcos0
d t d t

Wu
r d2Q r dQ

R'X sin(0+e)+ — R'X( — )« cos(0+e)«0

g dt* g dt

If we assume R large, for an elementary dis-
placement, R dQ may be considered rectangular,

hence the term...

wr dQ

— R'X(-r~)* cos(0+e) may be omitted.

g dt

Further R * R1 appro*, R being the distance from
axis 0 to the trunnion.

In experiments, conducted by the French at
"Sevran-Livry" the term 2mrw -T— was found to be
negligible. Hence the equation of angular
motion about the trunnion axis without an
elevating gear interposed becomes,

n T \dw Wr ^aQ
T r sinal**'1tr+1tc'^fjr -WrXcos0 RX — T sin(0+e)»0

since Trsina1 and WrXcos£l are small for a large
angular acceleration, we have, approximately,

vdw *r

From this equation we observe that immediate-
ly upon the recoiling parts becoming out of
battery, when the acceleration of the top
carriage is backwards, as would occur in the jump
of a field carriage, the upward rolling of a ship
or in the recoil of the top carriage in a double
recoil system or railway mount, we have an angular
acceleration tending to cause a reversal or stress
in the elevating gear.

878

ANGULAR ACCIL1RAT16N OP THI TIPPING PARTS.

Invariable Elevating Gear Reaction
Introduced.

In this case, tbe angular acceleration of tbe
tipping parts, is the same as the angular ac-
celeration of the system about the fixed or in-
stantaneous axis 0. To impress this angular ac-
celeration on the tipping parts, as would occur in
the jump of a field carriage, or in the upward
rolling of a ship, tbe elevating gear reaction is
lessened or completely reversed when tbe trunnions
are located along tbe bore. Considering fig. (3)

Let Pfc = the powder reaction on tbe breecb (Ibs)
Qt and Qa = tbe front and rear clip reactions
(Ibs)

tan u = f = coefficient of guide friction
mr and wr = mass and weight of recoiling parts
(Ibs)

8 = total braking force (Ibs)

X and Y = components of tbe trunnion reaction

parallel and normal to tbe bore (Ibs)

E = elevating gear reaction (Ibs)

j = distance from trunnion axis to line of

action of E (ft)

9e = angle between E and tbe axis of tbe bore
7r = relative velocity of recoiling parts in
cradle (ft/seo)

dQ

— = angular velocity impressed on tipping parts

dt (rad/sec)

Ir - moment of inertia of recoiling parts about

center of gravity of recoiling parts
Itr moment of inertia of recoiling parts about

trunnion axis.
Itc = moment of inertia of the cradle about

the trunnion axis.
xo and yo = battery coordinates of the center of

gravity of the cradle with respect to the

879

Fig. 3

880

truonioD axis,

x and yt = battery coordinates of the center
of gravity of the cradle with
respect to the trunnion axis.
dtb - distance from trunnion axis to line of

action of B.

r' = radius of the trunnion bearing,
u = friction angle in the trunnion bearing.
x y and x y = coordinates of the front

and rear clip reactions
with respect to the

trunnions.

BBACTIOM6 OH THE RECOILING PARTS.

The reactions on the recoiling parts are:

(1) The powder force — Pb (Ibs)

(2) The reactions due to the con-
straint of the cradle — Q and Q2

(Ibs)

(3) The braking force exerted by the
cradle — B (Ibs)

(4) The relative tangential inertia

force due to the relative acceleration
dv of the recoiling parts —
mr -^ — (Ibs)

(5) The relative complementary
centrifugal force due to the com-
bined angular and relative motion of
the recoiling parts

dQ
2mr vr — (Ibs)

d* (6) The tangential inertia force due

t. rotation a"bout the axis 0

m.R (Ibs)

dta

(7) The centrifugal inertia force
due to rotation about the axis 0- —

»rR(^)2 (Ibs)

(8) The weight of the recoiling
parts Wr (Ibs)

881

(9) The angular couple resisting

d*Q
angular acceleration Ir—-j (ft.lbs)

d *

Tbe equations of notion for the recoiling
parts, become, along x x1 —

avr d*Q dQ

pb-ar — mr R -777- cos(e+0)-MrR(T-) sin(e+0)+Wrsin0

since u = 0 (1)

along v v '

dQ ,/dGN, r, d'Q
rv_ -— +mrR(— --)zcos(e+0)-mrR -—
dt at at

(2)

for moments about the axis 0, we have,

dvr

r dt

. d«Q . d«Q dQ .

-ffi-R* 1_ +2m_vr — [xx-x+Rsin(£J+e)]

dt9 dt» dt

-Wr[ (x0-x)cos0+Rsin e-

Now mrR2+Ir=Ior moment of inertia of recoil-
ing parts about axis 0.
Hence the above expression reduces to,

(Pb-mr^)[Rcos(e+0)+s]+Pbeb-ZW0(Qt + Q2+B)-Ior j^~
a t

dQ

+2mr»r — [x0-x+Rsin(0+e)]-Wr[^0-x)cos£!+Rsin e-yosin0]
d t

(3)

BBACTIOHS OH THB CRAPLB;

The reactions on the cradle are:

U) Tbe reactions of the recoiling
parts on the cradle Q4Qa and E.

(2) The trunnion reaction T = /X2+Y2

and having a moment about its
center line Tr1 sin u.

882

(3; The elevating gear reaction E

(4) The tangential inertia force

. R *I£

dt* da ,

(5) The centrifugal inertia ra«(— )2

c
dt

(6) Tbe weight of the cradle Wc
The equations of motion, become

along the x x1 axis

(Q. +Qa)sin u +B+»Tcsin0-t-Ecos 9

d*Q .dQ

— cos(e+0)-mcR(—

d*Q dQ a

-mcR — cos(e+0)-mcR(— ) sin(e+0)-X =0 (I1)

along the y y1 axis

Y+Esine -W cosD-CQ -Q )cos u-nn_R(— )a cos(e+0) (21)

d t

for moments about the axis 0,

d*0
2M0(Qt*Q2*B)-XR cos(0+e)+YR sin(0+e )-mcRa -

dt2

d*Q
-Iv— > +Ecos6e[R cos(0+e)-Jcos9e]+Esin9e[Rsin(e+0)-J

sin9e)-Wc(Rsin e-xccos0+ycsin0)=0
Now,«cRa+Ic=Ioc the moment of inertia about the

axis 0 of the cradle.
and Ecos9etRcos(0+e)-Jcosee]-»-Esinee[Rsin(0+e)-Jsin Qe]

=ER cose cos(0+e)+ER sin0esin(0+e)-EJ(co3aee+sin*ee)

=ERcos(0+e-9e)-BJ=E[Rcos(£)+e-ee)-J]

Hence the moment equation of the cradle about 0,

reduces to ZM (Q +Q +B)-XRcos (0+e) + YR sin(0l+e)-Ioc— -

dt*

+E[Rcos(0+e-ee)-J]-Wc(R3ine-xccos0+ycsin£J)=0 (3 ' )

HgACTIONS ON THE TIPPIMQ PASTS.

Since the tipping parts are balanced about the

883

trunnions in the battery position, we have,
and

*rxo~wcxc*° •rxo~moxc=0

Adding (1) and (I1), (2) and (21) and (3) and (3»),
we have

d*Q dd t

(Hr+Wc)sin0+Eeos8e-X=0 (1")

dQ jn -

Y+Esine«»-(Wr+W.. )cos0+2mrvr-— + (mr+nu
c r c r r j * * r c

d*Q
-(•r-t-mc)R— — • sin(0+e)=0

dvr

Cl If M W V_l U

[x0+x+RCZJ+e)]+Wrxcos0-(Wr+Wc)Rsin e-XRcos (0+e)

+YR sin(0+e) + E [Rcos(0+e^-9e)-J] =0 (3")

Equations (1"), (2"), (3") are the general
equations of a recoiling system, where the
relative translation is along the axis of the
bore and the trunnions have a rotation about some
fixed axis 0.

These equations may be simplified as fol-
lows:

W^aWr+Wc ana mt=iBr+mc where Wt= the total

weight of the
tipping parts.
mt= the total mass

of the tipping
parts
Further Ior»Itr+mrR*

Ioc*rtc*"cR* and Ior+Ioc=tItr+Itc+BtRt
Rcos (0+e)+s*Rcos(0«-e) approx.

dvr d2Q dQ

Xapw~mr mt Rt cos(0+e) + ( — )*

b dt dt* dt

884

d*Q dQ dQ

[ - sin(0+e)-( — )*cos (j0+e)]-2mpvr— ' -Esin0ft
dt« dt dt

*Dd d«Q dvr

-(Pb-ar—

Rcos(0+e-ee)-J
sin ee-Wpxcos0+HtRsin e+XRcos(0+e)-Y in(0+e)

1R)

Substituting the values of X and Y in the equation
of £ and simplifying, wa have,

J

(4)

which is evidently the moment equation of the
various kinetic reactions on the tipping parts
about the trunnion as an axis. Since the term

dQ
2mrvr— (x0-x) is always small, the elevating

^* dv gear reaction, reducesato

Itr + Itc)— (4')

where Itr*Ir+«r[ (xo-x)8+y«l

Ir= JDoment of inertia about center of gravity
of recoiling parts. Hsnce I^r is a
variable depending upon the displace-
ment in the recoil x, also Itc=Ic*mc ^xc+y

a constant
Ic= moment of inertia about center of

gravity of ths cradle.

The equation (4) or (41) is of special im-

portance in the study of the variation of the
elevation gear reaction. The angular acceleration

be detericined in the following discussion

886

on the jump of a carriage.

In the case when s and eb = 0, that is when the

center of gravity of the recoiling parts and trun-
nion axis lie along the axis of the bore, we have

-(Ti +T* ) - Thus the elevating gear reaction

dt8

is reversed and its monent

about the trunnion imparts the
required angular momentum in the tipping parts.

We calculate the value of (-E) we must determine

d2Q

the maximum angular acceleration — -— .

d t

The condition that there will be no reversal
of stress on the elevating gear on the jump of
a field carriage, is that

Wrxcos0+(Pb-mr— )s+Pbeb rc

dvr
Now roughly Pb~flr "~ ~~~ ^ tne static resist-

dt

ance to recoil, hence for no reversal of stress
on the elevating gear,

In the battery position:

Ks+Pbeb ^ (itr+itc

Cut of battery position:

>/ .d2Q
cos 0 =(It.r + It.«

From these equations we may determine the
required distance from the center of gravity
of the recoil parts to the trunnion axis, to
prevent a reversal of stress on the elevating
gear when the gun jumps as in a field carriage.

RiCTILIKBAR ACCELERATION OF THE
TIPPINS PASTS

With a double recoil system, or in the case
of a railway recoiling along the rails, the trunnions
are accelerated to the rear due to the recoil re-

886

Fig. A-

887

action of the gun. Thus the tipping parts are sub-
jected to a rectilinear acceleration to the rear
and the elevating gear reactions is increased.
Considering fig.( 4) we have the various re-
actions as the recoiling parts and cradle as
shown.

BKACTIOHS OH THE BgCOILIHQ PARTS.

The reactions on the recoiling parts, consist,

(1) The powder force -- Pb (Ibs)

(2) The reactions due to the constraint
of the cradle --- Ot and Qf (Ibs)

(3) The braking force exerted by the
cradle — B (Ibs)

(4) The weight of the recoiling parts —
Wr (Ibs)

(5) The kinetic reaction of the recoil-
ing parts due to the relative accel-

eration — dv_

«r— - (Ibs)

(6) The kinetic reaction due to the ac-
celeration <Jy

(— — ) of the constraint of

d* the recoiling parts,
i.e. the top carriage and cradle —

dvc

•r — (lb"

Then, for the kinetic equilibrium of the

recoiling parts, along the axis of the bore

dvr dvc

pb-or -- "i-— cos(»+»)+Wrsin«J-(Q1+0J[)sin u - B=0
dt dt

along the normal axis to the bore,

dvc

(Qz-Qi)cos u-Wrcos0-mr - sin(0+a)=0 (2)

d t

and for moments about the trunnion axis,

u

dvr dvc
-mr[ - + - cosd^+a^lttQ-WpCos 0(xQ-x)

u v u. v

888
d?

r— - sin(0+a)(x0-x)+Wrsin0y0 =0 (3)

dt

For the kinetic equilibrium of the cradle,
ire have, along the axis of the bore or guides,
(Qt+Qa)sin u+E cos QQ+B-mQ—~ cos (0+a)+Wcsin0-X=0

along the normal to the guides or bore,

dvc
Y+Esin8a~(Q2"~Qt )cos u -Wccos0— mc— r— sin (fif + a)=0 (21)

and for moments about the axis of the trunnions,

dvc
it a t COS1 iyt eyz s b mc (jtcos ra yc

dvc
+mc— - xcsin(0+a)+wccos0xc-Wcsin0yc-Ej = 0 (31)

dt

REACTIONS ON THE TIPPING PARTS.

Since the tipping parts are usually balanced
about the trunnions in the battery position, we

have,

wrxo~wcxcs° "r^c" mcxc =0

and

Adding equations (1) and (I1), (2) and (2'), (3)
and (3'), we have, then, along the axis of the

bore,dVr dvc

Ph-m_ -- (m_+m_)— - cos(0+a) +(W,+W_ )sin0+Ecos 0a-
" dt " c dt

d")

along the normal to the axis of the bore,

dvc
Y+Esin9Q-(Wr+Wc)cos0-(mr-nnc)— - -sin(0*a)»0 (2")

dt
and for moments about the trunnion,

dv dv

c

.— • sin((2f+a).x-Ej=0
The elevating gear reaction, becomes,

889

dvr d»e

(Pb-mr-—-)y0+Pbab+Wrx cos0+mr--— sin(Gf+a).x

d t d t

From equation (1),
dvr dvc

PK-BP[— rr -- TTCOs(0+a)] +Wrsin0-(0 +Q, )sin u-B=0
dt at * a

Since the displacement and velocity of the
top carriage is small at the beginning of recoil,
the relation vr= the static velocity vs, that is
the velocity of the recoiling parts when the top
carriage is assumed stationary. cv*

The braking force B, equals, B=F?+-^j—

"x
but since vr-vg approx.

cv£ cv£

B=F» »ag"sF..+ T^ ~ B- where B_= the static re-
v wx v w*

coil braking force.

Hence the kinetic reaction along the bore, becomes,

dvr dvc
Pb-mr[ I cos(0-»a)] = BS + (Q +Q )sin u -Wrsin0

CL if Cl ti

But for the static resistance to recoil, we have

)sin u-Wrsin

hence

dv, dv,

X C

Pb-mr[ - + - cos(0+a)]=Kx Therefore, the

dt dt elevating gear re-

action reduces to,

dv

This equation is of special interest since
in the battery position, we find,

dvc
pb«b *Kxy0+mr— y0cos(2>+a)

E= when the top car-

J riage moves.

0

when the top carriage is stationary,

J
Thus we have only a slightly additional load

890

mr—

brought on the elevating gear (Ibs)

J
This value however is somewhat compensated

by the slightly decreased value of Kx due to the
fact that the relative velocity is somewhat less
than the static velocity of recoil.

RECAPITULATION.

Reaction of Top Carriage on Tipping Parts:
For tbe Trunnion Reactions

dvr dvc

=Pv-inr-- -- (mP+nic)---rcos(0+a)+(Wr-»-Wc)sin0+Bcos9e
0 * dt dt

dvc
m+m) - sin(0+a)-£ sin 9

dt

For the Elevating Gear Reaction
dvr dvc

in (0+a)

J
If we define Kx=B+(Qt+Q2)sin u+Wrsin0

then dvr dvc

Kx~pbsi"r[ - + - cos(0+a)] and Wt=Wr+Wc total

* weight
of the tipping parts, Mt=mr=mc Total mass of

tipping parts.
For the Trunnion Reactions

dvc
X=Kx=mc-— — cos(0+a)+Wtsin0-«-Ecos9e

dvc

Y sW^cos^+mt-— sin(0+a)-B sin9.»
at c

For tbe Elevating Gear Reaction:

dvc
Pbeb*Kxyo+wrx cosef+mr— [xsin(8f+a)+yocos (0+a)]

_ ___^_ __ dt

E=^ - '

J

ON THE JUMP OF A FIELD CARRIAGE Mounts are

frequently de-

signed for stability at a given minimum elevation
and yet may be fired at a lower elevation. Con-

891

sideration, therefore, must be given to the inertia
loadings and corresponding reactions induced by the
jump of the carriage. ID the following discussion
it will be assumed the total mount to rotate about
its spade point.

By the application of D1 Alerabert 's principle
we introduce the various inertia effects as kinetic
reactions, the mutual reactions between the parts,
of course having no effect on the kinetic equilibrium
of the total system, gun cradle and carriage.

From the acceleration diagram we have for the
recoiling parts,

(1) The relative acceleration along the
axis of the bore --

dvr

— Ut/sec.«)
dt

(2) The tangential acceleration of'tbe
recoiling parts about the axis 0 -

BS=

dt

(3) The centripetal acceleration of the
recoiling parts towards the axis 0 —
w«R

(4) The acceleration due to the relative
motion combined with the rotation

of the recoiling parts 2w vr

The accelerations in the remainder of the mount,
the carriage proper, become

(1) The tangential acceleration — kc""~~

dt

(2) The centripetal acceleration — w2Lc

KIHSTIC EQUILIBRIUM OP THE SYSTEM.

(Gun and Carriage)

Prom the principle of D'Alenbert, we have the
external reactions in equilibrium with the various
kinetic reactions induced by the angular rotation
of the mount and the relative acceleration of the
gun.

The forces and kinetic reactions on the system

892

gun and carriage are :

(1) The total powder reaction P^

(2) The weights of the recoiling
parts and carriage W^ and Wc (Ibs)

(3) The tangential inertia force of
the recoiling parts due to the
angular acceleration about the

spade point 0

dw
MRR — (Ibs)

(4) The centrifugal inertia force of
the recoiling parts due to the
angular velocity about the spade
point 0

MR R w2 (Ibs)

(5) The inertia resistance due to the
relatire acceleration of the recoil-
ing parts

(Ibs)

dvr

r dt

(6) The inertia resistance due to
the combined rotation of the
recoiling parts

2 mrwvr (Ibs)

(7) The tangential inertia force

of the carnage proper due to

the angular acceleration about the

spade point 0

dw . .
fflc c dt

(8) The centrifugal inertia force due
to the angular velocity about the
spade point 0 mo Lcw* (Ibs)

(9) The inertia couple about the
center of gravity of the recoiling
parts due to the angular acceleration
of the system —

IT? TT (ft. Ibs)

893

(10) The inertia couple about the
center of gravity of the car-
riage proper due to the angular
acceleration of the system —

*£$*$• 'oJi (fulbs) <r»«'

For moments about the axis 0, we have,

dvi

dw

P (d+e)-(DR d-mRR

dt dt

(x0-x)-Wp[(x0-x)cos0-d sin0]
dw -a dw dw

*4t smcLc dt c dt c c cos <e-B> - 0(1 )

n
since 9 = -% +0+Q whence Q = angle turned in rotating

about 0, we have ^Q ^

— = — = w for the angular
dt dt

velocity

dae daQ dw

-^— = = — for the angular acceleration.

dt* dt2 dt

Considering now the recoiling parts, above,

we have

d v t? d w
pb~aR mR d — -mR«»*(x0-x)-B-Rt+Wrsine(=0.

dt dt

R dw
Simplifying we have, Pb^Rt-jT + d T^-«-w^c0-x)]-B-Rt

+Wrsin(?=0 (2)

whence B=Fv+Pb — tne total braking reaction (Ibs)
Py = the recuperator reaction (Ibs)

Pn - the total hydraulic resistance (Ibs)
Now v*

pb=phs ~ whence

vg = static recoil velocity (ft/sec)

Pns = corresponding static hydraulic braking
reaction (Ibs)

894

a
dvR (jw VR

We thus see that Pb~IBRtT7-fdTrtw* (xo~x)33phs ~T~*Rt

dt ut v

Fv-WRsin0 (3)

From equation (1), we have

dvp

Pb(d+e)-mR - d+2mRw vR(xo-x)-WR[ (xQ-x)cos0-d
d« _ _ dt

dt " mRR*-mcL0=IR+Ic

sin£J]-WcLc cos(0-B)

If Is3 the moment of inertia of the system about
the axis 0, then Is»mRR*+mcLc+IR + Ic a variable

since mRR*«mR[da+(xo-x )a] a function of x, hence

dvR
pb(d+e)~"mR~d"[ d+2mRw vR(xo-x)-WR[ (xo-x)cos(Z(-d

dt 5 Is

sin]-W.Lrcos(6-B)

- — - (4a)

Substituting the value — in equation (3), we have

dt dvR

a dynamical equation in terms of — — — and w. If

dt

now. we construct a table for the various intervals

of time, we may compute VR, p , w and J* by the

d t dt

methods of a point by point procedure.

APPROXIMATE CALCULATIONS FOR THE Prom equation
JUMP OF A CARRIAGE (3) in the

previous article,
we have

dvR dw VR

Pb-mR[ - +d. — +v»2 (x0-x)]-Phs — -vFv+R
dt dt v

—

Cl tr

The terms m d— and mR w2 (XQ-X) are usually

small compared with dvR

dvi

mR , hence VR=VS approx

dt

and Pb~mR~j~ = ^ ^^e statlc resistance to recoil

(approx )

895

COMPONfA/TS OW JUMP

/WffTM SOffCfS ON JUMP OF F/fLD

Fig. 5

886

Substituting in equation (4a) of the previous article,
and omitting the term 2 mr«r vr(xQ-x) which is small,
we have
dw Pbeb+Kd-Wr[(x0-X)cos 0-d sin0]-W0Lc cos(e-B)

The moment effect of the weights,
Hp[(x0-x)cos0-d sin£J)= WcLccos (9-B)= wsLs~BR
Hence dw Pbeb=Kd-WgLg +WRx cos 0

** " "V
where Ws = weight of entire system

Ls = horizontal distance from spade point to

line of action, of Ws
Is = moment of inertia of total system about

spade axis

BARBETTE CHASSIS MOUNTS. In this type of mount,

the top carriage and gun

recoil up an inclined
plane, and the recoil
in general is not parallel
to the bore.

The characteristics of such mounts is that a
component of the direct powder reaction is brought
upon the mount and therefore the various parts are
stressed considerably higher then with mounts recoil-
ing in a cradle. During the powder period, we have
an impulsive or percussion effect brought on to the
mount, and the effect of finite forces as gravity
and the braking force may be neglected.

The gun together with the top carriage are
considered in this type of mount as the recoil-
ing parts. The gun has trunnions, and the trunnions
are located at the center of gravity of the gun
along the axis of the bore. Since there is no
regular acceleration in the recoil, the reaction
on the elevating gear is practically nil. Due to
the weight and position of the center of gravity

897

ON fffCO/L/A/G

Fig. 6

898

of the top carriage, the center of gravity of the
recoiling parts is not located at the axis of the
bore. During the powder pressure period, there-
fore, we have a whipping action due to the powder
pressure couple which increases the end roller re-
action and the front clip reaction.

The bottom carriage which supports the
chassis for the top carriage is traversed on a roller
base plate, the horizontal reaction being carried
on the pintle bearing and the vertical reactions
by the traversing rollers. This arrangement is
typical of any Barbette emplacement. Let
Fb - the total powder reaction (Ibs)
0 = angle of elevation of gun
a = angle of inclination of chassis
fflg and Wg = mass and weight of the gun (Ibs)
DC and wc 3 mass and weight of the top carriage
mr and wr - mass and weight of the recoiling

parts

8 = the total braking reaction (Ibs)
dt>= distance from trunnion to line of action

of B (ft)

QA and 0^ = the front and rear roller reactions
on the top carriage exerted by
the chassis (Ibs)

Rt and Rf = the front clip reaction and rear
roller reaction exerted by the
traversing base plate on the bottom
carriage (Ibs)
dx and dt = distance from trunnions to line

of action of Q and Q respectively,
n = friction angle of roller reactions.

H - the horizontal reaction between the base
plate and bottom carriage at the pintle
bearing (Ibs)

REACTIONS OtT THE RBCOILIHG PARTS

GUN AMD TOP CARRIAGE T06STHBR

We have for the motion of the recoiling parts,
along the chassis:-

899

d*l

Pbcos(0+a)-W_sin a-B(Q +Q )sin u-m. — =• =0 (1)

rdt»

normal to the chassis:-

PhSin(0+a)+Wrcos a-(Qt+Qt)cos a =0 (2)

about the trunnions:—
d*l

If we assume the braking constant throughout the
recoil, we have, B+Wrsina+(Qi+Qa )sin u = K and
equation (1) becomes,

Pbcos(0+a)-mr— • - K =0

Integrating, we find,

dl Pbcos(0+a) K

— = v = / - dt -- t
dt m m

= Vf or the maximum free

r r

velocity of recoil for
a recoiling mass mr, hence

Kt

t
dt

Integrating again, T Kp

1. * / Vfcos(0+a)dt- —
o 2mr

KT2

= E cos(0+a) - — • where E is the free recoil dis-
2nir

placement for a recoiling mass
m. during the total powder period. During tfte re-

g

mainder of the recoil, we have J»rv*=K(b-lt ) hence

rr t f^ flpg

-mr[Vfcos(0+a) -- ]* *K[b-Ecos(0+a)+ — ]
mr 2mr

Simplifying, mfV| cos«(cr+a)

K = - where E and T
2[b-(E-VfT)cos(0+a)] are obtained

by the methods of Interior Ballistics.

EFFECT Of CHASSIS ROLLER REACTIONS OH THE RKCOIL
BRAKE.

Assuming only the end roller reactions to come
into play, we have, from eq. (1) and (2),

900

K-B-Wrsin a

tan u = • = f

Pbsin(0+a)+Wrcos a

hence K -B-Wrsina=f [Pbsin(0+a)+Wrcos a] where f »
coefficient of roller friction. After the powder
period, K-B-Wrsin a = f W*r cos a, therefore during
the powder period, B1=K-Wrsin8J-f [ Pbsin(0+a)+Wrcosa]

in the recoil, B=K-Wrsin0-fWrcos a , and the charge
of required braking, becomes, B-Bt» fPbsin(0+a)

BEACTION3 ON THE BOTTOM CARRIAGE.

The reactions on the bottom carnage are:-

(1) Qt and Q2 reversed, the roller
reactions on the chassis of the
top carriage (Ibs)

(2) V reversed, the braking reaction

(Ibs)

« t-

(3) The horizontal pintle bearing

reaction n.

(4) The weight of the bottom carriage

Wtc-

(5) The traversing roller and clip
reactions Rt and R8 (Ibs)

Then, resolving forces along and normal to the
chassis, we have,

(Qt+Qa)sin u +B-HCOS a+ (Rft-Rt )sin a-Wtcsin a =0 (I1)

(Q +Q )cos u+Wt.cos a-Hsin a -(R -R )cos a =0 (21)
a t wO a i

and for moments about the trunnion,

H dh-Bd^Q^-Q^-R^-R^-W,^ =0 (3 • )

where x^ = the momentum of Wtc about the trunnion.

BXTSRNAL REACTIONS ON THE SYSTEM CONSIST-
I NG OP1 THS TOTAL MOUNT.

Adding equations (1) and (I1), we have,

dal
Fbcos(0+a)-Wrsin a-mr-— • -Hcos a+(Ra~Rt)sin a-Wtc

sin a « 0 (1")

901

Since Pbcos(0+a)-mr — pK and »r+Wtc=Ws the total
d t

weight of the mount. Equation (I11) reduces to,

K-Hssin a - H cos a +(Ra-Rt)sin a = 0 (1")

Adding (2) and (2'), we have

Pbsin(0+a)+Hrcos a+Wtccos a-Hsin a -(R2-R1)cos a »0

(2")
Adding (3) and (31), we have,

- (y0cos a+x0sina)»0 (3B)
dt*

Eliminating (Ra~Rt) from (1") and (2"), we have

Kcos0-H+Pbsin a sin(0+a) =0 (a)

Eliminating H from (1") and (2"),

(R2-Rt)+Ksin a-Pbsin(0+a)cos a-Ws=0 (b)

and equation (3") reduces to for moments about the
trunnion,

sina)=0 (c)

From (a) and (b), we have, H=Kcos0+Pbsina sin(0+a)
R>-R1»Pbsin(!^+a)cos a+»s-K sin a. Substituting the

value of H in (c) and combining with (b) we obtain
R( and Rt respectively.

PERCUSSION REACTIONS:

The percussion reactions take place during the
powder period and are reactions of a magnitude
comparable with the powder forces. In an ordinary
cradle recoil, the direct effect of the powder re-
actions are practically eliminated by allowing
the gun to recoil along the bore. In mounts of
the chassis type, especially when the gun elevates,
we have a large component of the powder reaction,
which causes the chassis to offer a corresponding
reaction.

902

OA/

Fig. 7

903

FERCUSStQH REACT/ON
ON RECO/L/N6 PARTS

PERCUSSION REACT/ ON ON TOP CARRIAGE

*J

t \

Fig. 8

904

In dealing with impulsive forces, the effect

of continuous or finite forces is negligible com-
pared with the percussion reactions.

Hence in the following we will omit such forces
as gravity, and the recoil brake reaction.

PERCUSSION REACTIONS ON THE RECOILIN3 PAHTS:

The percussion reactions are,

(1) The powder force — Pb

dal

(2) The inertia resistance I=ffi_— — •

dt2

(3) The resultant reaction of the
chassis --- Q

P^ acts along the bore, I acts parallel to the

chassis and through the center of gravity of the
recoiling parts, while Q. balances these reactions

at their intersection, as shown in fig.(8)«
The force polygon of the percussions is abc,
where a b //P^, bc//I and ca//Q. The direction of
Q. is slightly inclined to the chassis due to the
friction angle u. Further Q is the resultant of
Q and 0 the front and rear roller reactions. Now

t 2

the resultant of Q and I, must intersect the
resultant of P^ and Q8 . Since P^ intersects at
02 at a, we have the direction of the resultant
of Q, and I along ae . In the force polygon bd is
drawn parallel to ae, and therefore cd is proportion-
al to Ql while da is proportional to Qa .

In the force polygon, we have,

Pb I Qt Q2 vc .

— = — = — = — hence I = — P^,

ab be cd da ab

""^*

DYNAMICAL RELATIONS ON FIRING Small guns up to
FROM AN AEROPLANE. a caliber of 75

have been successfully fired from large aeroplanes,

8/

905

Larger calibers may be possible by the introduction
of the muzzle brake, which thereby reduces the re-
coil reaction.

In this discussion, however, we will take the
simple case of a gun without a muzzle brake. Let
VQ = horizontal velocity of the plane before

firing (ft/sec) Vo

V a velocity of the plane immediately after
firing (ft/sec) Vt

Vr = velocity of the gun at the end of the

powder period (ft/sec) V^
v = muzzle velocity of projectile (absolute)

(ft/sec)
Pb - powder reaction (Ibs)

R = recoil reaction (Ibs)

inr and wr = mass and weight of recoiling

parts (Ibs)
ng and *s ~ mass and weight of equivalent

weight of aeroplane + weight of
cradle and mount (Ibs)

Assume the gun to be fired horizontally while the
aeroplane flies horizontally:

During the powder period, we have the mutual
impulsive reaction between the gun and aeroplane =
Fb dt t

For the gun, / * Pb dt = mr(Vo-Vr) (1)
o

the impulsive effect of the recoil reaction R being
negligible. For the projectile and powder, we
have, Ato

f Pb dt =(m+0.5m)(v-V0) during the travel

up the bore.

N 7~Vo

Pb dt = mU700-(— ) during the powder

° expansion.

906

At

/ Pb dt * I(v-V0)+i4700 * mv+m4700 (approx)

0 (2)

Let us DON consider the effect of the recoil
reaction R on the aeroplane and fixed part of the
mount. On firing the aeroplane the aeroplane acts
somewhat as an elastic beam, more or less supported
by the air reactions at the ends. We may consider,
the equivalent mass of the aeroplane and mount at-
tached = 0.7 to 0.8 the actual mass of the plane
and mount. We will denote ms as this equivalent
mass.

Then, for the motion of the aeroplane during

the recoil period, we have,
«S(V0-V )

R = S ° f Ubs) (3)

t

and for the motion of the recoiling parts during
this same period,

nr(V -Vr)

R = * (IDs) (4)

t

since the recoiling parts must have the same
velocity as the plane at the end of recoil.

It is interesting to note the magnitude of the
relation of the various velocities for a typical
small mount.

V0 = 100 miles/hour = 146.6 ft/sec.

V0-Vr» 30 ft/sec, roughly; Vr = 116 ft. sec. roughly,

V = between 116 and 146 ft/sec, say 130 ft/sec.
Thus vie have a check in the velocity of the plane
of several feet per second, the magnitude of which
depends of course on the ballistics and relations
of the various masses.

Combining the previous equations, we have,

•r(V0-Vr)=mv+i 4700 (5) '

(Wg+«r)(V0-Vt)«mv +1 4700 (7)

That is, as we should expect from first principles,

907

the momentum imparted to the aeroplane backwards,
equals the momentum imparted to the projectile
and powder forwards.

Let us now assume the recoil reaction con-
stant, and let b equal the length of recoil.

Now due to the superior motion of the aero-
plane as compared with that of the gun, during
the recoil the aeroplane does work on the gun, in
bringing the velocity from the smaller value Vr to
the larger value V hence

2
The energy taken from the aeroplane, becomes

2

hence -R b = -| (V*-V») - -|(V0-?«) therefore,

the recoil

reaction, becomes _ v« »8 +

_ lr ^"o ""rvr. xms*mr.ri,

R%-[ (-r * — >-(-T— )v; '

(Ibs)
.hare Vr=Vo -(211i|225, ({t/sec)

aod vvo-(HIiZ22i) ((t/seo)
ms+mr

DISAPPBARISG AHP OTHBH TYPgS OP CARBIA6E8.

TYPES OP DISAPPEARING Disappearing gun car-
CAFRIAGES. riages, as evident by their

terminology, are designed,

so that in the recoil the gun is brought down below
a parapet and disappears from the enemy's view.

The gun is loaded in the lower position. By in-
troducing a counterweight, the gun is brought by

gravity to the firing position, the gun during
the firing period only being above the parapet.
Disappearing gun carriages may be broadly

908

classified in two general types:-

(1) Revolving or rotating types,
where the gun lever rotates about
a fixed axis, as in the Monorieff.
Howell and Krupp carriages.

(2) Sliding carriage types, where
the Cardon system of linkage is
used, the gun lever being constrained
to move at two of its points along
guides practically at right angles,
as in the Buffington Crozier models.

APPROXIMATE THEORY OF THE The following as-
ROTATING TYPE OF DISAPPEAR- sumptions are made and
ING CARRIAGE. the validity of these

assumptions will be
considered more in detail later:-

(1) The center of gravity of the gun
will be assumed at the gun trunnioni

(2) The angular displacement of the
gun lever, during the powder period,
will be assumed small and will

therefore not effect the initial
geometrical conditions greatly.

(3) The inertia effect of the elevat-
ing rods, will be assumed negligible
as compared with that of the gun,
lever, gun and counterweight.

(4) The elevating arm, will be assumed
approximately parallel to the axis
of the gun lever and roughly equal
to the upper half of the gun lever.

(5) The angular movement of the gun
itself during the powder period will
be assumed very small.

From assumptions (3), (4) and (5) we may
neglect the reaction of the elevating arm during
the powder action period, for the following reasons:
(a) The tangential component of

909

the elevating arm reaction
becomes zero due to assumption
(3).

(b) Condition (4) assumes the
instantaneous center of the
gun practically at infinity.
Hence the angular velocity of
the gun at the end of the
powder period is negligible;
the angular acceleration
therefore may be assumed zero,
and the normal reaction of the
elevating arm becomes zero.

In practice it is possible to obtain (1) com-
pletely, and (2) and (3) are closely realized. The
condition (4) may be met constructively at one
elevation but is difficult to meet for all elevations,
since the gun customarily is designed to recoil to
the same loading angle.

To reduce the reaction on the elevating arm
it is customary to introduce a kick down buffer at
the bottom end of the arm, and thus during the
powder period a small minor reaction comparable with
the buffer resistance is introduced between the
elevating arm and gun. This reaction may be neglected
as compared with the major reactions of the gun
lever.

Therefore, as a first approximation, however,
we will neglect the reaction of the elevating arm,
and assume the center of gravity of the gun located
at the trunnions. Let

Wg= weight of the gun (Ibs)

Wr= weight of the gun lever (Ibs)

wcw= weight of the counterweight (Ibs)

Ir=WrkJ = moment of inertia of gun lever

about fixed axis of rotation.
Icw = Wcwk§w = moment of inertia of counter-
weight about fixed axis of
rotation.

910

REACTIONS ON THE. ROCKEIR AT GUN
DURING POWDER PERIOD

REACTIONS ON THE ROCKER AT GUN
AFTER POWDER PERIOD

w

cw

Fig. 9

911

T and N = tangential and normal trunnion re-

action (Ibs)

X and Y = horizontal and vertical reactions

at axis of rotation of gun lever (Ibs)

P * total powder reaction (Ibs)

Pffl = maximum powder reaction (Ibs)

0 = angle of elevation of gun
6^= initial angle of gun lever with respect

vertical
0£= final angle of gun lever with respect vertical

r = radius of upper half of gun lever (ft)
r1 = radius to center of gravity of counter-

weight (ft)

R = reaction of oscillating cylinder brake
d^= initial angle R makes with the normal to r'
ro = distance from axis along r1 to line of

action of R.
m = mass of projectile
IE = mass of powder charge

v = muzzle velocity (ft/sec)

TfVf = total friction torque resisting rotation

From fig.( 9). the gun axis makes an agle 0- 9j
with the tangent of the path in the initial position
of the gun.

For the motion of tte gun lever, we have for
moments about the fixed axis,

+* cos d • ro + Vf (1)

and for the motion of t"he gun along the tangent to
its initial path, „

Pcos(?- 6i)-T- -* r — (2)
g dt»

If s = the displacement along the arc of the gun

trunnion

V = the corresponding tangential velocity of
the gun trunnions,

912

ds d6 d*s dV d*e

— * r — ; • = r - Hence, combining

dt dt dta dt dt* the two equations,

we have,

wg *r *cw dV ro rf

Pcos(0-6.j )=( — +— -+- r—)— +Rcosd; — +Tf — (3)
1 g r2 ra dt r • r

Evidently — J+~"£"~ may be regarded as the so called

equivalent translatory mass at the

gun trunnions due to the rotational inertia effect
of the gun lever and counterweight.

Integrating equation (3), we have,

r0 rf

Rcosd — dt Tf— dt

Pcos(0-6i) r c r

V = / — — dt - / -— /

g Ir Icw Wg Ir Icw Wg Ir Icw

^P I -f-_^__ " I HM I ---- -,_S \ \ --

g r2 r2 g r2 r2 g r* r2

Now both 0 and d as well as the friction torque
TfTf vary during the powder period but as the
change is small, we are quite justified in assum-
ing them constant. Further, since, Pdt=(m+0.5m)v
(during the travel of the shot up the bore), we
have

(m+0.5m)v
— - - dt = - - — - - cos (0-°i) or in terms

Wrf 1 r i /% n- W/< 1— -l-riui

8 , r , cw (_5+_j, cw\ of the free

g r2 ra g r2 r2 velocity of

recoil, (m+0.5m)v

Vfcos(0-e.)= - - — - - cos (f?-9i)

W

wr xr 1cw.

( — + — + - )

g r2 r2'

where Vj is the equivalent free velocity with a

recoiling mass equal to

"r lr xcws

( — + — + - )

g r2 r2'

Integrating again, we have

Xfeos(0-ei)=["*0'5mi — ]x' 008(0-6^ where x1 = the

g ^ r | cw absolute dis-

& r placement of
the projectile up the bore. Now

913

•-*)
x'cos(0-6i)= u cosdy-QiJ-XfCosCCf-Si) hence, we have

wg lr Jcw

— + — + - +m+0.5m

g r2 r2

Ir Icw

now m+0.5m is small compared with — +-T+-— r bence

g r2 r2

we may assume

(m+0.5i)ucos(0-ei)
Xfcos(gf-6i) = - - — - - i — (ft)

The equations of recoil, become therefore

^2+Tf — )t
r rr

V=VfCos((?-9i)

and

Ic,

'"r1

(Rcosd— +Tf — )t*
r Ar

(approx)

(m+0.5m)v (m+0.5m) u
where Vf= and X

wr ^ Jcw , ,wg Jr ^w.
( — +—+ — r) ( — +-T+ — r) + m+ 0.5 m
g r2 r2 g r2 r2

We see the equations of recoil during the
powder period are exactly similar to the previous
recoil equation, the recoiling mass now including
the inertia effect of the rotating elements. Hence
the previous interior ballistic formulas are im-
mediately applicable for the computation of the
free recoil displacement E and the time of the
powder period te.

For the maximum velocity of recoil, we have

mv +47001
Vfm = " — (ft/sec) and the max.

^-£+:£?L velocity of

& r r constrained

recoil along the path of the gun trunnion, becomes,

914

(Rcosd— +Tf—)tc
r r

IS+T —
r r

The corresponding maximum angular velocities
and angular displacements, become,

,de. V" s.

"«*(dI)m=T and e* " T

The energy of recoil at the end of the powder
period becomes, w

A s i(T +T + — i r9 )n9
Hm a v r xcw TT m

From the energy equation we may easily consider
the remainder of the recoil.

Since the brake and friction resistances are
small compared with the powder reaction and
the inertia resistance of the rotating parts, we
may assume with sufficient accuracy that

vm=vfmcos^0~9i^ and sm 3 B cos(*-0i) We have,
for the recoil energy at any angular displacement
6.

f (Rcos d.r0)d9 + / ' Tf rfde+Wcwr ' (cos Qj- cos 6)
0i 9i

- Wgr r(cos Qj - cos 9)= Am- A where Wgr = weight

of gun and that

portion of the rocker, not including the counter
weight reduced to an equivalent weight at the gun
trunnion, that is

T" r" = u Wg where r' = radius of

915

Wrrr
*gr=Wg+ ' rr = distance

from axis to

center of gravity of rocker. Since d varies with
the angular displacement of the gun lever, from a

layout we may readily evaluate the term

8
f (Rccs d rQ)de provided R is assumed constant

i which is usually the case.

Further since T^r£ does not vary greatly we may
assume it constant. As a close approximation,

u(Wg+Wr+Wcw)rf* u =0.15 roughly

radii

bearing of
axis of rotation of rocker

r" = radius of trunnion. Further T*rf=T Irf+Tflr"
f f f f f

Hence 9*

/ Tfrf de = Tfrf (6f- 6i) (ft/lbs)

now A = r(Ir + Icw+ — ra)ws
2

(rad/sec)
REACTIONS OK THE CORDAN LINKAGE Reactions on the

DISAPPEARING CARRIAGE DURING Gun: The center

THE POWDER PERIOD. of gravity of the

gun is assumed at
the trunnion axis

of the gun. The angular acceleration of the gun
is assumed small and the reaction of the elevating
arm on the gun is considered a secondary force, this
being possible by a proper arrangement of the parts
or by the introduction of a kick down buffer at the
base of the elevating arm.

The primary reactions on the gun consist:

(1) The powder force along the axis
of the bore = Pb

916

f?£ACT/OA/S ON T/if GUN

.

Fig. 10

917

(2) The trunnion reactions divided
into horizontal and vertical com-
ponents X and Y respectively.

(3) The weight of the gun acting
through the trunnion axis = Wg

(4) The tangential inertia force
along the path of the movement
of the trunnions or normal to a
line from instantaneous axis to

the trunnion axis =

d2s

• nirf

*dt*

(5) The centrifugal inertia force,
normal to the path of the trunnion
axis and proportional to the square
of the angular velocity = dQ

sftp*

The secondary reactions on the gun are:

(1) The elevation arm reaction on the
gun comparable with the kick down
buffer reaction at the base of the
elevating arm.

(2) The inertia couple due to the
angular acceleration of the gun
about the trunnion axis. .

In the following analysis, we will neglect
the effect of the secondary reactions. The forces
on the gun neglecting the secondary forces are
shown in fig . (

Since we assume the rotation negligible, we
have the equations of motion,

Pbcos 0-mg — cos B+mgl(-— -)2 sinB-X=0

T-mg^| sinB-mgk(~)2 cos B+Wg-Y =0

b d2s d*6

where tan B = — - tan 6 ; — -= 1— -r approx. since
a+b dt1* dtz

1 does not

change greatly during the powder period.

918

ON TH£ GUN

FACTIONS ON TflF SUD//V6

Fig. I

919

I = / (a2 + 2ab)cos26+ba Hence, the trunnion re-
actions become,

d28 d6
IT— rcosB+m,* 1 (~)2 sinB

at s at

Y - Pvsinef-D-1— —sin8-mcl(-— -)2cosB+Wtf
5 dt * dt

BUCTIOMS OK THE BOCKKB.

The reactions on the rocker, are:

(1) The reaction of the gun on the
rocker divided into components X and
Y.

(2) The reaction of tlie sliding car-

riage on the rocker at the rocker
trunnion, divided into components

X1 and Y1.

(3) The reaction of the counterweight
cross head at the wrist pin of ttie
cross head, divided into components
X" and Y".

(4) The weight of the rocker at the
center of gravity assumed at the
rocker trunnion Wr.

(5) The rotational inertia couple
due to the angular acceleration of
the rocker = ,2

Ird~t7

(6) The tangential inertia force of
the rocker along

d*x
OX * mrI7jr actinfi

through

center of gravity of rocker.
(7) Tbe centrifugal inertia force
of the rocker normal to
OX

at

The equations of motion of tte rocker, become,
along OX -

Y-Yi+Y"-m — — =Q

x * A " °

920

~fri

along OY

about the instantaneous axis I,

X(a+b)cos 9+Y b sin 0-X'a cos 9-Y" a sin 9 - mr

dfx d29

_ a cos e - lr _= o

BBACTIOHS 01 THB SLIDIHG CARRIAGE AHD
COOMTBR WEIGHT RB3PBCTIVSLY IN

THB DIRECTION OF THBIB MOTIONS.

Considering the sliding carriage, we have,

d*x

X1— H-mc— — -=0 Where R is the hydraulic brake re-
action en the carriage and for the

counterweight,

d y

Y"-mcw w =0

dt2

EQUATION OF MOTION OF THE SYSTEM DURING
THE POWDER PRESSURE PERIOD.

Substituting Vhe values of X',Y",X and Y in
the moment equation about the instaneous axis of
the rocker, we havs,
Pb I ( a + b )

cos e cos 0+bsin 9]-m,. [ (a+b )2cosa6+b2sin29]

* dt2

+Wgb sin 9-[R*(mr+mc) ]a cos 9 mcw — —• a sin 9

* dt2 dt

-Wcw a sin 9 - lf 1J[ = 0

Now x = a sin 9

dx d6

— = a cos 9 —
dt dt

d2x d29 He,,

jpj- = a cos 9 — — a sin 9 (-^)2

921

and y * a cos 9

dy d9

— ~ - a sin 9 —
dt dt

d*y d*9 de,9

_ = - a Sln 0 _ - a cos 6 (_)•

If we assume the positive direction of y upward,
then

Substituting these values in the arova
equation we have the general dynamical equation
of the disappearing carriage during the powder
pressure in terms of a single coordinate variable

e.

The differential equation of motion, becomes,
Pb[(a+b)cos 9 cos(8+b sin 6 sin 0]-nig[ (a+b)*cos* 9+t>»

j 2 ft j j A

sin8 9] - •'•W.jb sin 9-B a cos 9-(mr+mc)a2cos* 9 -
dt* dt*

d29 do

-m_wa2 sin2 6 — - +(m..-»-m..)a sin 9 cos 9( — )* -mrw
dt2 dt

a sin 9 cos 9(~)« -Wcw a sin 9 - I — *0
dt dt*

Combining terras, we have, Pb[(a+b)cos 9 cos0+b sin 9

sin0]- <ing t(a+b)2 cos2 9+b2 sin2 9]4(mP+mc)a2 cos2

1 d29
e+mcwa2 sin2 9 + 1 I+t (mc^pJa sin 9 cos 9 - mcw

d0
a sin 9 cos 9] ( — )a -R a cos 9+W,, b sin 9 -Wcwa sin

dt

0=0

The equation is in the form of A— — +B( — )2+C=0

dt2 dt

where

922

A=«g[(a+b)acosa 6+b«sina

sin 6 +Ir

B= -[(mc+nr)a sin 9 cos 6 ™Cwa s*n 9 cos ®^

C= -Pb[ (a+b)cos 6 cos tf+b sin 8 sin 0]+Ra cos 6-Wg

b sin 6 +WCW a sin 6

CALCULATIOM Of THE RECOIL DUBIHG THE
POWDBR PRESSURE PERIOD.

The general equation of motion for the system,
becomes,

pb[(a+b)cos 6 cos0+b sin0 sin0]-[mg(a+b )* cos* 8
+b2 sin2 6] +S;TDr+mc)a2 cos8 6 +mcw a2 sin8 8+Ir]

— - +[(mc+nr)a sin 6 cos 8 -mcvf a sin 8 cos 8] (—

Q V d tr

-R a cos e+*gb sin 8-Wcw a sin 8=0
We nay urite this, as

APv-B - +C( — )2-D =0 where A =(a+b )cos8cos0+bsin8sinCf
dt2 dt

B »e»g[ (a+b)acosa8+hasin8]
+ [ (mr+nic ) a2cos8+mc|f a2

C = (mc-*-»r)a sin6cos6-mC)|a sin8cos8
D =Ra cosSHfb sin e + Wa sin 8

Integrating, we have

t J/\ t

now f Pjjdt =(n+0.5l)v during the travel up the bore

where m = mass of the projectile

a = mass of the powder charge

v = velocity of the projectile in the bore.
hence

923

OF

y FOX

* INSTANTANEOUS C£NT£# OF &OTAT/ON

. 12

924

-7 s |(m+0.5m)v+ - J (-7)* dt - - t which is the

at o D Q ut V\\ B

general ex-
pression of the angular velocity of the system during

the travel up the bore. Integrating again,

6= -(m+0.5m)u+- / / (-7)* dt.dt - •£- t* where u»

0 D A n (It GO

the travel
up the bore

These equations may be integrated by a point by point

method .

KINEMATICS OF A CORDAN LINKAGE In the analysis
DISAPPEARING CARRIAGE INCLUDING of the kinematics
EFFECT OF ELEVATING ARM. of any linkage,

we have two sys-
tems of diagrams,

viz:- velocity diagrams and acceleration diagrams.
By the use of the velocity diagram, we may calculate
the centripetal accelerations, which of course must
be included in the acceleration diagrams. Due to
the required velocity diagram, we are justified
in using the instantaneous center about which the
gun lever rotates.

Therefore, in considering the instantaneous
center, the cordan linkage including the elevating
arm, becomes, a four bar linkage and we may con-
struct a velocity diagram as for any four bar

linkage. The acceleration diagram of the linkage,
however, is not theoretically equivalent to a four
bar linkage, since the instantaneous center of
the gun lever, has a definite path in the recoil.
Hence the distance from the Instantaneous center
of the gun trunnion, changes in ths recoil and the
tangential acceleration becomes,

d (wl) dlA , dw

• *» - «• 1 - Since 1 does not charge great-

dt dt ly during the powder period

we are justified in omitting

925

dl d(wl) dw d«9

w —- being small then — — * 1— i —

dt dt dt dt*

Consider any position of the linkage during the

powder period: see fig. 03).
Let

9 * angle made by gun lever or rocker with

vertical (rad)
1 = distance from instantaneous center to

gun trunnion (ft)

B = angle made by 1 with vertical (rad)
d = distance from gun trunnion to elevating

arm trunnion on gun (ft)
0 = angle d or axis of bore makes with

horizontal

C = length of elevating arm
Q. = angle made by c with vertical (rad)
x0 and y0 = coordinates of base of elevat-
ing arm (ft)

dfl

w = — = angular velocity of gun lever (rad/sec)
dt

dQ
wi - — = angular velocity of elevating arm

VELOCITY DIAGRAM.

The linear velocity of point 0, becomes,

dt

The component along "d" becomes, Iw cos(#-B)
Tbe linear velocity of point Q1, becomes, cw1

Its component along d, becomes, cw1 cos(0-ft)

Hence Iw cos(0-B)= cw ' cos(0-a)

1 eos(0-B) dO 1 cos(0-B) de
and "' '' c cos(0-Q) " '"• dt " c cos(0-Q) dt
The angular rotation about the trunnions, equals,

926

FOX PO/NT O,

. wc-
oo,

13

827

d0 _ Vel.O-Vel.O _ H'C sin(0-Q)-wl sin(0-B)
dt " d d

... cos (£)-B) _, rt. ,,., Ov,w

-Q)-l sin(0-B)]-

(rt *\
cos(0-Q) d

7 [tan(0-Q)cos(0-B)-sin(0-B)](4^) (2)

d dt

ACCELERATION DIAGRAM:

The acceleration of point 0, the center of
gravity of the gun, along the x axis,

d*x d*8 ,d8,

_ g , 1 — cos B - 1<-)2 sin B

Along the y axis,

dtz dt2 dt

Tbese values do not include the effect of the small
change in "1" in the powder period. To include this,
we have Xg=(a+b)sin 9

dxg d9

— — =(a+b)cos 8 —

dt dt

=(a+b) cos 8 - -(a+b)sin e* (3)
dt dt

and y^= - b cos Q

dyg de

- = b sin 9 —
dt dt

j r

= b sin e+b C08 e ()» (4)

dt* dt8 dt
The acceleration of 0', is divided into the follow-

ing components:

(1) The acceleration of 0, divided
into two components
daxg d«vg
dt2 dt2

928

(2) The centripetal acceleration
of O1, about 0 directed along d
towards 0 *

dt

(3) The tangential acceleration of
0', about 0, normal to d and equal

dta

Since 0', is a common point for both the gun and
the elevating arm, we have, also, the acceleration

of O1, divided into,

(1) The tangential acceleration of

the gun lever at 0
d«Q dw'

(2) The centripetal acceleration of
the gun lever at 0

- c(— )* * c*1*
dt

"From the acceleration diagram, we have the
following vector equation

dt« dt» dt dta dt* dt

The two unknowns in the equation, are

d*Q
^— and c - which we will denote by ad and ac

dt» dta

bat we have two coordinate equations

along ox and oy and hence a solution is possible.

The solution may either graphical or analytical,
He have, from (3) and (4)

.(a.b)t=o,9 - sin e()
at» dt1 dt

929

!•(->

d (—)**[ tan (0-Q ) co s(0-B)-sin (0-8)]* - — —
•It i

,dQ » ,« 1« cos«(0-B) d6.»

c( — ) -cw1 = - - ( - ) From the ac-

dt C COS8(0-Q) dt celeration

fU J.C e* fce*»? rj**,",* -rj ••fc?'? t • .. ceieranon

diagram, we
have along the x axis.

d'yg ^djO.a ,dQ,a

r~z -- drrr) cos0+adsin0=a,,cos Q-c ( — ) sinQ
at* at at

,N .,

d^dT slnjZJ~adcos^*acsln®+c^!J — ^ cos Q

then,

2 dQ

acsindcosQ=— -5— sinQ-dCrr) sinQcos0+adsinQsin0+c (-— )
ct t d L d t

sin2Q

acsinQcosQ= cosQ-df-—-) sin0cosQ-ajcos0cosQ-cf~)

dt2 dt dt

cos2Q

ROTATING TYPE CARRIAGE: The maximum reactions
REACTIONS ON TRUNNION on trunnions and main
AND FIXED AXIS OF bearing (fixed axis of rocker)
ROCKER. are at a maximum value at

the maximum powder pres-
sure, and therefore we only need to consider these
values in determining the strength of parts. The
powder reaction is mainly balanced by ths insrtia
resistance offered by the gun and the revolving
parts. The reaction exerted on the rocker at the
trunnions, is that needed to overcome the angular
inertia of the rocker and counterweight which in
turn must be equal to the powder reaction increases
the inertia resistance offered by the gun. There-
fore the heavier the gun as compared with the re—

930

volving parts, the smaller the effect of the powder
reaction.

The reaction of the main bearing is consider-
ably augmented over that of the trunnion reactions
due to the tangential inertia forces of the counter
weight. The development of the Cordan linkage in
which the rocker bearing is allowed to slide back
on a top carriage has been largely to decrease the
reaction at the main bearing when fixed as in the
revolving type.

At the maximum powder force, the recoil
velocity of the gun is small and therefore the
centrifugal force of the gun may be negleoted. The
tangential component of the trunnion reaction, be-

T=Pjnaxcos(0-ei) + Wsin 9^ — where

mjjar*

and for the normal component, N=Praaxsin(0-6 j )+WgCos
Therefore, we have

T=proaxcos(ei-ei)(i -- -i - ]+wg

*r *cw
in** — + -
1 r« r»

N=Pmaxsin(0-ei)+Wgcos Bi _
and for the resultant trunnion reaction S^ s / N2+T*

The maximum bending moment in the rocker or
gun lever occurs at a section adjacent to the center
bearing of the rocker. This bending moment is due
to the moment of the reaction of the gun at the
trunnion minus the inertia moment of that part of
the rocker above the section, which is practically
one-half the mass of the rocker or gun lever.

The moment of the inertia resistance of the

rocker, becomes,

- s

t d26 l . dta i lr d«s lr .
- \- - = - I_ - = - — r - where — is the
r dt2 r « r« dt2 r*

equivalent mass of the gun lever referred to the

931

trunnions. The maximum bending moment at center
section of the rocker or gun lever, becomes,

t Xr d*s,
M0=vT - - — - - — -)r or in terms of the maximum

w * r at

powder force
Ir

Ar xcw
no +—•+——
« r» r»

In addition the section is subjected to a
compression, CO=N+-^ Wrcos ei=PBaxsin(0-9i ) + (Wg*^Wr)

We will now consider the reaction at the fixed axis
of the rocker or gun lever. Since the tangential
inertia effect of the rocker practically balances,
we will consider the reaction on the main center
bearing as due only to the reaction of the gun at

the trunnions and tne inertia of the counterweight.
The tangential inertia resistance of the counter-
weight, is

d«e

dt*

Fcwslmq. - — — where q is the distance

to any mass particle of
the counterweight measured from the axis of rotation
of the gun lever. If rcw = the distance to the
center of gravity of the counterweight, then

Imq»McwrCB hence

d26 , . a .rcw Mcw
FCW=MCW rcw Jj#*.wftrtQF*i>-J —

It is to be noted that the point of application
of Fcw is not at the center of gravity of the
counterweight, but rather at the center of percussion
of the counterweight with respect to the axis of
rotation of the gun lever. If k is the distance
from the axis of rotation to the center of per-
cussion, (J2Q

where Z - the
/* + * A + a

radius of gyration

of the Counterweight with respect to the fixed axis,

cw Z2

therefore k = - Resolving" the resultant

rc» reactions at the fixed

932

axu of the gun lever into components normal and

alop~ the axis of ths gun lever, we have, neglect-
ing the centrifugal forces as small,

Y=N+(*cw+Wr)cos ®£ or substituting values for

N, T, and Fcw,

r

cw

X=Pjnaxcos(0-9i)(H- - ~-^ - )+(Wg+Wcw+Wr)sin 6i

Mcw — • ^

Y*pmas

and for the resultant we have, SQ

From these equations, it is easy to see, that the
reaction at the fixed axis is increased over that
at the gun trunnions by the tangential inertia of
the counterweight ^ P^cos (0-6 A

r " " IM

With a heavy counterweight, this term is larger
and the bearing load at the fixed axis becomes
very great with large guns. To reduce this re-
action and consequent weight of members, etc.,
the Cordan linkage disappearing carriage developed
by Buffington.Crozier and the Krupp linkage
have been used for the larger gun mounts. Sub-
tracting, we have,

-—^ sinQ- r-r^cos Q+d(—)*sin(0-Q) +adcos (0-Q)t-c(-— )
dta dt2 dt dt

d*xg d2yg

Substituting the values of - and - , we have

dt2 dt2

(a+b)[cos 6 - sin 6 (^)«]sinQ-b[ sine ll|+cos(li)t

dt* dt dtz dt

cosQ+d(~)asin(0-Q) +adcos(0-Q)+c $-)* =0
dt dt

Expanding and simplifying, we find

933

d*0 d6 2 da0

a[cos6 — - - sin 6(--) ] sinQ-b[sin(6-Q) — -

at at dt2

dt

,dQ.« I2 cosM0-B) ,d0xa
and I—*) = — • — — — — (— • )
dt c* cos*(0— Q) dt

hence

-:»ni?f Jsap* sdj

ad

b[sin(e-Q)— +co«<^Q)(-j7)*]-a[cose S-t - sin
dt at dt-

cos(gf-Q)

()] sin Q * --

-*-t-h

'ft*' »

... ^_pk t 55- V c

;-'l^ cos8 (0-8)1 d6 ,

sin(0-Q)+ — ,' ^. > (— )

c cos2(0-B)J dt

f -*. ,-: :

Combining the acceleration and velocity terms, we
have

[bsin(9-Q)-a cosSsind] d26 <{a sin Qsin Q+b
a , * — — + >

cos(0-Q) dt* cos(0-Q)

f*f^>d« Jliii ^(iSJSo --— -• ^ i)

i* i

cos(e-Q) - — rtan(0-ft)cos(0-B)-sin(0-B)] sin(0-Q)>

AX *

d9 « 1^ cos2(0-B) d9 2
dt c cos2 (0-B) dt

Therefore the angular acceleration of the gun, be-
comes,

A —
ad d dt2'

dt2 d d cos(0 - Q)

(rad/sec.2 )

834

where Ad» b sin (9 - Q) - a cos 9 sin Q

Bd - a sin 9 sin Q+bcos(6-Q) [tan(0-Q)

d

cos(0-B)-sin(0-B))asin(£J-Q) + — cos2(£KB)

c cos*(0-Q)
d Q
For the acceleration ac = c -^— we eliminate arf in

dt*
the equations:-

dtt.cos£H3(— )au dt<

-accosQcos0

d*yg d0 a dQ 2

= TTT- sin0-D(-~) sin*0-c( — ) cosQsinfl
at dt dt

-acsinQsin(?

Adding, we have

,d0 2 dQ a

in0-d(— ) 0-c(— ) sin(0-Q)-accos(0-Q)=0
dt* at* dt dt

hence
*c 3 cos(0-Q)

g g a a

Substituting for , 0 , , ., (--— ) and (—— ) we obtain
dtz dt2 dt dt

dt2 dt dt* dt

cos(0-Q)

cos(0-Q)

eosg(0-Q) dt

935

Combining, we have, r i

« /y u /*. /*\ 4bsin (6-0)+acos0sin9+-

acos6eos0+bcos(e-0) d*e I d

cos(0-Q) dt»

[tan(0-Q)cos(0-B)-sin(0-B)]*+—

cos(0-Q)

c cosa(0-Q)J

dt

d*Q
Therefore since ac = c , the angular acceleration

dt* of the gun lever,
becomes, 2

d*Q 5 dt a

i 160I.-3V

d c cos(0-Q)

where A., = a cos 9 cos 0 + b cos (6-0)

r

= - •sbsin(9-0) + a cos0sin9+-— [ tan(0-B)-sin

c cos* (0-0)

RECAPITULATION 07 VELOCITIES AND

ACCBLBRATIOHS IH A CORDAH LIHKAGB
DISAPPEABIN6 GUN CARRIAGE:

Let a+b = total length of gun lever

a = distance from cross bead to top carriage

trunnion
b = distance from top carriage trunnion to

gun trunnion
d = distance from gun trunnion to elevating

arm trunnion measured along gun.
c = length of elevating arm.
3 = angle gun lever makes witn vertical
0 = angle turned by gun
Q = angle elevating arm makes with vertical

M

— = angular velocity of gun lever

dt

936

£ = angular velocity of gun
at

dQ

— » angular velocity of elevating arm

d«9

•— • = angular acceleration of gun lever

d»0

TTT z angular acceleration of gun

d*Q

777- s angular acceleration of elevating arm

dxg

-— • = horizontal linear velocity of gun at
trunnions

- — = vertical linear velocity of gun at
trunnions

a = horizontal acceleration of gun at trun-
nions

, s = vertical acceleration of gun at trunnions

at

dx

— = velocity of top carriage
at

d»x

-— - * acceleration of top carriage

dt*

dy

— * velocity of counterweight and crosshead
at

d»y

TT7 = acceleration of counterweight and cross-

u t

bead

Then, in terms of the angular velocity and acceleration
of the gun lever,

(a) The velocity and acceleration

of top carriage, are
dx d9

- - • «». e - (ft/iee)

d*x da8 dfi 2

- = a cos 6 -- a sin Q( — ) Cft/seca)

dta dtz dt

937

0>) The velocity and acceleration
of tbe top carriage, are

" ~ a sin 9 —• (ft/aec)

at

= - a sin 9 ~j - a cos 6 (2)' (ft/sec«)

(c) The velocity and acceleration
of the gun, are

—^ = (a+b)cos 9(4r

dt at

- = (a+b)cos 0 ^Ii2_

dyg

—
at

d*y
-—
at

de

b sin 6 —-
at

d9 s

= b sin e —T+b cos 6 (-— )
atz dt

i[tan(0-Q)cos(0-B)-sin(0-B)]—
d dt

(ft/sec)

(ft/sec«)

(ft/sec)

(ft/sec«)

(rad/sec)

df

(rad/sec«)

where A = b sin(9-Q)-a cos 9 sin

B = a sin 9 sin Q + b cos(9-Q) -- [tan(0-ft)

cos(0-B) - sin(0-B)]in(e>-Q) +

c cos2 (

(d) The velocity and acceleration
of tbe elevating arm, are

dQ 1 cos(g-B) d9

= (rad/sec)

cdt

dt2 c cos(CT-O)

where A_ = a cos 9 cos 0 + b cos (9-0)

12

Bc = ~ "J b sin(8-0) + a cos $ sin 9 + — [tan(0-Q)
L c

..a 1* cos2(0-B)l

cos(0-B)- sin(0-B)J + — «/* »\ r

« c cos2 (0-Q) J

Coordinates of the system:

Displacement of top carriage = x
Displacement of counterweight = y

Distance from instantaneous center of gun
lever to gun trunnion:

1 = Aa+b)2cos2 6+b2sin2 6

COORDINATES OP THE CORDAN LINKAGE In estimating
DISAPPEARING CARRIAGE. the work done by

the various weights
and resistances
during the retard-
ation period of the recoil it is necessary to com-
pute the various displacements of the parts of the
system in terms of the independent coordinate
of the system.

Prom the diagram, to determine VI and Q in
terms of the angle 9 made by the gun lever with the
vertical, we have

xo = (a+b)sin 6+d cos 0-c sin Q

y0 » - b cos © + d sin 0 + c cos
which may be written,

d cos 0= xQ-(a+b) sin 6+c sin Q

d sin 0=yo + b cos 6-c cos Q
Squaring and adding, we have d*=[xo-(a+b )sin 9]"

+2[x0-(a+b)sin 9]c sin Q +(yo+b cos 9 )%2(y0+bcos 9)

c cos Q + c*

This equation may be put in the form,

939

[xQ-(a+b)sin 9]sin Q+(y0+bcos 9)cos Q
[x0-(a+b)sin 9)2+(yo+bcos 9)2

2

hence m sin (A+Q) = J "S da7C2 + [xo-(a+b)sin 9)%(yo+bcos9 )*>
where m * /[xQ-(a+b )sin 9]2+(yo+bcos 9)a

_. xft-(a+b)sin 9
A = tan [- °

yo+bcos t

From this equation we may solve for Q in terms of
9, and substituting in either equation below,

cos 0 = 3- [x0-(a+b)sin 9+c sin a]

a

sin 0=5 lyo+k cos ® ~ c cos ^

we may then calculate the value of 0 in terms of

the independent variable. Further if,
Displacement of top carriage - x
Displacement of counterweight = y

The distance from instantaneous center of
gun lever to the gun trunnion, is

1 = /(a+b)acos2 9 +ba sin2 6

8EACTION3 ON THE PARTS OF Considering the re-
CORDAN LINKAGE. actions on the gun, it

will be assumed that
ths center of gravity
is located at the gun

trunnion. The gun is subjected to a translatory
acceleration divided into horizontal and vertical
components as well as an angular acceleration due
to the reaction of the elevating arm. Let

(1) PI, = the powder pressure along

the axis of the bore

(2) X and Y = the horizontal and

vertical reactions at the

gun trunnions.

940

(3) W s the weight of the gun acting

through the gun trunnion

(4) nu*-* and md * ? = the inertia

8 dta

components

along the horizontal and
vertical axis

(5) *gT~7 * the iner*ia angular re-

sistance

(6) X"'and Y'"= the horizontal and

vertical components
exerted by the ele-
vating arm on the gun

Then for the motion of the gun, we have
daxg

- -— - x"'= 0
c d t

Pbsin0-Y-m-

dt«

yMIdco80-X'"d

dt*

For the elevating arm, we have (X1 ' 'cosQ+Y1 ' 'sinQ)c-

d«Q

Ic — - = 0
5 dt2

where Ic = the moment of inertia about its fixed

axis.

Combining with the moment equation of the gun, we
have

cos 0"1* d~F c sin Q

(Ibs)

cd cos(0-Q)

d2Q
Ic— d sin0*Ig — c cos 0

Y"'= - (Ibs)
cd cos (0-a)

Next, to obtain the reactions X and Y we must consider
the dynamical equations of the gun lever. By taking

moments about the instantaneous center of the gun

941

lever, we eliminate the unknown normal reactions
of the constraints of the carriage and counter-
weight.

Then for moments about the instantaneous
center of the gun lever, we have
X(a+b)cos 9+Y b sin 9-X'a cos 9~Y"a sin 8 - ra

d'x d*9 d'x

— a cos 6 . IR — = o where X'-R*mc—

R s the hydraulic

brake reaction
on the carriage "mc".

day

Y"»mcw — — +wcw mcw and wcw » mass at

dt*

weight

of the counterweight. Combining, we have the
dynamical equation of the motion of the disappearing
gun carriage during the powder pressure period, as

follows:

d«Q d*0
— — •dcosfl-I-— — csinQ
cdta Sdta

- - - ^]
cd cos (0-Q)

c cos

cos 9

[Pbsin 0-mc -- J b sin 9

dt8 cd cos(fr-d)

dax day

-(R+ra - ) a cos 9 - (mcw - + wcw) a sin 9

dt« dt«

d»9

- raB - a cos e - ID - * 0 *

dt* dt2

For a solution of this equation we must
substitute for the various accelerations their
value in terms of a function of tne acceleration

d*6

- . The hydraulic brake resistance R may readily

dt* be obtained by considering the energy equation

942

of the linkage to its recoiled position.

If Ag 3 kinetic energy of gun at end of powder

period (ft/lbs)
Ac » Kinetic energy of top carriage at end

of powder period (ft/lbs)
Ae = kinetic energy of elevating gun at end

of powder period (ft/lbs)
AR * kinetic energy of gun lever at end of

powder period (ft/lbs)
Aw ~ kinetic energy of counterweight at end of

powder period

Then for the kinetic energy of the gun, we have,
if 1 - the distance to gun trunnion from the

instantaneous center of gun movement, and
k radius of gyration about center of
gravity or trunnions of the gun.

2 )

d J dt

(ft/lbs)
For the kinetic energy of the elevating arm.

••-% BOO •*$ }

» _ 1* eos*(0-B) d6

c2 cosa(e-Q) dt
For the kinetic energy of the gun lever, if kE »

* NOTE: If the path of the sliding carriage

has an inclination to tha horizontal equal to angle

d, then for the equation of tho gun lever, we have

a2*

X(a+b)coe O +Y(b sin O - a cos 0 tan d ) - ( R + m c - )

dt2

Substituting the values of X and Y as in the
previous equations, we have the general dyna»ioal
equation of Motion.

943

radius of gyration about the center of gravity of
tbe gun lever, we have

A

R

If tbe top carriage and sides are inclined plane
making angle « with tbe horizontal.

'

9(l+tana V./TB.B.

dt

For tbe kinetic energy of the top carriage
Ac=rl"ca*cos8 e^TT^ for horizontal plane and

Ae = 7 meaacos2 9(l+tan* o)( — )* for inclined plane

dt

For the kinetic energy of the counterweight and
cross bead, t

Aw = I mwa2 sin* 9("cTt^a

When tbe sliding carriage rides an inclined plane
the kinetic energy of the counterweight, becomes,

1 2 d 0 2

Aw = -fflwaa(sin 9+cos 9 tan <*) (— *-)
From the principle of energy,

where WB ~ work resisted by the recoil brake

WCB = work resisted by the weight of the

counterweight
Wg = work done by the weight of the gun

We = work done by the weight of the elevating

_ arm

WR = work done by the weight of the gun lever

Wc = work done by the weight of the sliding
carriage

During the powder period, the sliding carriage moves

a distance E and tbe gun lever angle increases from
60 to 9j . The length of recoil = L and the re-

coilad position of the gun lever makes an angle 6
with the vertical.

944

Work, resisted by the recoil brake » WB jf R=the
brake resistance, then for the work of the recoil
brake during the retardation period, we have

WB » R(L-E) (ft/lbs) where obviously L-E»a(sin 62-3in
and with an inclined plane sliding carriage,

Work resisted by the weight of the counterweight * W

cw

"cw ~ *w yw "w * "eight of counterweight
where yw = a(cos 9t-cos 9a) and with an inclined
plane sliding carriage yw = a(cos 6t-cos 6a)+L sin a
Work due to the weight of ths gun = Wg

Wg= "g^g "here yg = (a+b)(cos 9t-cos &2) and like-
wise with an inclined plane sliding carriage
yg=(a+b)(cos 6t-cos 92)

Work due to the weight of the sliding carriage and

gun lever =» Wr+W,. ,f Assuming the center of gravity
of the gun lever at the gun lever trunnion, the
center of gravity of the gun lever has the sane
displacement as the sliding carriage. Hence

W.+W *(wr+wc)yc where ye » (L-E)sin a » — * — (sin6 -

1 ** COS **

sin BI). Hence when the plane is horizontal no
work is done by the weights of the gun lever or
sliding carriage.
Work due to the weight of the elevating arm = We

where ye = de (cos C^-cosQ,)

de = distance to center of

gravity from fixed axis
of elevating arm.

= sin

in-1 4

{dg-c»+[x0-(a+b)sin QI]'

2 /[x0-(a+b)sin eja+(yo+b cos

946
(a+b) sin 9

- tan" [-— - - - -]

yg+bcOsQj^

'

[{da-ca+[x0-(a+b)sin 8 l»+(y o+bcos 6
Qa » sin"1 «j -

L 2/[x0-(a+b)sin 9a 1» +(yQ+bcos 6^ )

xn-(a+b )sin 6

- tan"1 [-2— - g - •]

yo+bcos 92

EQUIVALENT MASS OF CORDAN LINKAGE. During the

powder period,
it is convenient

to express the dynamical equation of recoil in terms
of the external moments or forces and the equivalent
mass of the system tines the acceleration of the
coordinate considered. The equivalent mass and
corresponding reactions may be referred as a function
of the angle made by the gun lever witb the vertical
or as a function of the displacement of the slid-
ing carriage.

(1) Equivalent mass referred to angle
"6" of gun lever witb vertical :-

From the dynamical equation

of recoil for the Cordan linkage previously derived,
we have, for moments about the instantaneous center

of the gun lever, Phla cos 9 cos(9-0)]-Ra cos 9H0

*

a sin 9 » m- [ (a+b )cos 9 - £+ 0 8in 9 ft] +* a

dt» dt«

cos 9 — + mpw a sin 9 — — + I

dta dt« dt« c cos(0-Q)

daQ [bsin(9-Q)-a sin Q-cos 9] d20

dT2* +I^ d cos(Gf-O) dta

Neglecting the centrifugal components of the
accelerations, as small,

d*x a d«6 d*xg . . da9

- - a cos 9 - ; - a =(a+b) cos 9 -
dt« dt8 dt« dt*

946

day d*9 d*ytf d*9

» a sin 9 — — : • * b sin 9 — —

dt* dt* dt« dt*

d*Q 1 daO

TT * ,~ ^ ta c°s e cos 0+fa °os (9-^)]-— r

dt* c cos(Qf-Q) dt*

Substituting, we have Pj,la cos 9cos0+bcos (9-0)]-

Ra cos 9-ViLIBa sin 9 =

-a sinQcos9]*l
«cosa(0-Q) \

(a*+2ab)cos* 9+b*] +mca*cos*

[acos 9 cos 0+b eos(9-0)]« ^ [bsin(9-Q>-
cacos«(Br-Q) g

de _ d«9

— • Thus the equation is in the form of APh-BR-CW.=D-—
dt* dt

• here A = a cos 9 cos 0 + b cos ( 9 - (?)

B = a cos 9

C = a sin 9 and for the equivalent mass "D"

D * njg[(aa+2ab)cos* 9+ba ] +mca*cos*9+mcwa2sin*9+Ir

[a cos 9cos0+b cos-(9-0)]a [bsin(9-0)-asinQ

+ I * I

c*cos*(6-0) d2cos2(0-0)

cos 9]*

-^— — For the solution, during the powder

period, we express the powder re-
action as a function of the time, then

rlfl fl^ h DC\TOn«w A

UO m U x W « . A XX »f >

_./_«. (_ )t , _ MS -(_ ,t

where Vf is the velocity of free recoil of the gun

Integrating again, A BR+CWCW

a _ _ u TP _ f \ +2

• D ^ * ( - 20 )

where E is the displaceaent of the gun in free re-
coil. From the solution of these equations, we
obtain,

947

et*eo* Mge ~ <~ - >T* "here T = powder interval

BR+CW

The angular displacement and the angular velocity
of the gun at the end of the powder period. Sub-
stituting these values in the energy equation, we
have

*B * Ag*M*R +Ac+Aw+Wg+Jre+iR +*C-WCW and there-

fore can readily determine R the total braking
resistance.

(2) Equivalent mass referred to dis-

placement of sliding carriage X;-

In place of a movement and

angular acceleration equation, we nay consider
the inertia and the reactions of the system as re-
duced to an equivalent translatory mass and force
as a function of the displacement of the sliding
carriage. Therefore reducing the motion of the
system to one of translation along the path of
the top carriage.

By direct analysis, we have, if

Pjj * powder reaction

X and Y = components of gun trunnion reaction

XI and Y' = components of top carriage on gun

lever
X" and Y" = components of reaction on gun

lever at crosshead
X'11 and Y1 ' ' = components of elevating arm

reaction on gun
mg = mass of gun
mcw = mass of counterweight
T.S = mass of elevating arm

ro_ - mass of top carriage
c r •

mB = mass of gun lever

K e>

Then, for moments about the instantaneous center

948

d*x
X(a+b)cos 6+y b sin 6-Y" a sin 9-mR — a cos 6-Ir

dt«

-X' a cos 6 = o
dt»

Dividing through by a cos 9, we have

)y tan e-Y-tan e -.„

a 2 R dt* a cos9

Since

d20

sin 0

cd cos (0-G)

d20

cos

cd cos (0-Q)

ire have on substitution,

P^( - cos(?+-tan 6 sin0)-Wcwtan 6-X1 as the equi-

3 8

valent force acting along the top carriage guides

and _ d2Q J _ d»C

Ir— rr d cos0-Iflr— re sin

b

t

dta a *dt2 a cd cos

a sin JtfTa c cos v
a+b °dta gdt2 b . d*y

- - 9+ -

tan
a cd cos (fif-Q) ""dt*

Ir 4*9

+ — — — — — — as the equivalent inertia resistance

a cos9 dt» offered by the mass of the total

system reduced to the path along the top carriage
guides.

949

THROTTLING CALCULATIONS WITH ADD WITHOOT
A FILLING IM BPFF1B.

4.7 Gun Trailer Mount with U. S. Variable Recoil
Valve.

w = weight of projectile 45 Ibs.

v = muzzle velocity 2400 ft/sec,

v = 166.13 (in.)

w - weight of ponder charge 11 Ibs.
pb max * 34000 Ibs/sq.in.
b = 36" (0° to 45°)

4.7 (0° to 45°)

X = total resistance = 17806.9706 Ibs.

*r * weight of recoiling parts = 7560 Ibs.
St 3 spring load *

Pv£=Sf 3 spring load at end of recoil » 16140 Ibs
Fvj = So = spring load at assembled height =
Wr * 1.3 = 9800 Ibs.

16140-9800 6340
St * So " ~36 " "ST " 176-U1 lb8 "

increase of spring load per inch of
recoil.

« * maximum angle of elevation * 45 °
Wr sin a = weight component = 7550 * .70711.=

5338. Ibs.- 6805

Bg = stuffing box friction * 2,25 * 100 * 225
Rg = guide friction = Hru cos »
u = coefficient of friction « .15
Rg = 7550x.70711x.15 -800.8021

Effective area of recoil piston = 9.337 (sq.in)

950

METHOD OF PLOTTING VELOCITY CURVE - VARIABLE

RESISTANCE TO RECOIL.

Kt,

'P « Vfo - — ; (ft/sec)

2M

(ft)

When projectile leaves
the muzzle.

Kt.

(ft/sec)

2M

(ft)

The maximum restrained
recoil velocity and
corresponding recoil.

ob

where Vfm=Vfo * — (tm-to)[l- _^l__^_r_
Mr 4Mr(Vf-V0)

'ob

6Mr(Vf-V0)

](t.-t0) (ft)

and K(T-t0)

t. - T - -= — (Sec)

Kt
VP-Vf -- (ft/sec)

M
M

Kt*

Er>B -- (ft)
2Mr

> At the end of the powder
period

During the retardation period,
/ 2[K- \ (b+x-2Er)](b-x)

/ ^

and therefore

cA

2[K- - (b-x)-2Er)](b-x)

•

•(sq.in)

13.2

- Rt - W,

961

which gives the required throttling area with a
variable resistance to recoil during the retard-
ation period.

Sf-S0x

Ph=K+Wpsin0-Rt-(S0+ — ) for spring return re-
cuperators.

Equivalent throttling area : 4.7 A. A. Trailer, Model

1918
1_ 1 1

W2 \tf 2 U72

Area of one hole = .0113 sq. in.
In battery - Wxb= 20 holes = .226 sq.in. w| =.0510

Wx =103 holes =1.1639sq.in. w_ =1.3546
t * t

4" Recoil - W* =81 holes =.9153 sq.in. W* = .8377
xf xa

W« =88 holes=.9944 sq.in. W^= .9888

8" Recoil -Wx =86 holes =.8781 sq.in. WS * .9566

\ \

WXj = 77 holes = .8701 sq.in.W|t= .7570

12" Recoil -W, =98 holes=1.1074 sq.in. W. = 1.2263
xa *a

Wx =67 holes * .7571 sq.in. W| = .5732

i xt

16" Recoil -Wx =107 holes = 1.2091sq.in.W» = 1.4619

Wx =59 holes = .6667 sq.in. W» = .4444
*i *i

20" Recoil-W_ =115 boles = 1.2995 sq.in.W* « 1.6887

* 2 •

Wx =50 holes = .5650 sq.in. WJ = .3192
24" Recoil -W, =125 holes= 1.4125 sq.in.W* = 1.9951

X8 **

Wx = 30 boles = .3390 sq.in. W« = .1149

28" Recoil -Wx =140 boles - 1.5820 sq.in.W* -2.5027
WX2=30 holes = .3390 sq.in. WJ4= .1149

32M Recoil-Wx =140 holes = 1.5820 sq.in. W$ =2.5027
t

Wx = 24 holes = .2712 sq.in. W| = .0735
36" Recoil -Wy =152 boles=.1.7176 sq.in.Wx = 2.9501

*2 2

Wx =0 holes = 0 sq.in. W* = 0

962

Equivalent throttling areas-4.7 A.A. Trailer, Mpdel

1918

-1 - y, »• . i , y -*«-«
i»e y

In battery, i- •• — + — -=20.355, .049127, .221

W» .1.3546 .0510

4" Recoil,^- = — +— - — =2.204, .453720 .675

we .9888 .8377

8-B.oo.il.ij. 7^570 +i956e- 2.366 .422654 .650
12"Recoil,jL - -^ * j-^f™ 2.569, .389256. .623

6

16»8ecoil |j - -ijjj * -j-^ijj = 3.934 .340831 .683
20" R..011..JL. -ijg * T-i^ - 3.724 .2685S8, .818

11 1

24" Recoil,— = + * 9.204 .108648 ,329

we .1149 1.9951

28"Recoil,-— * — »4 = 9.102, .100865, .331

6 .1149 2.5027

32 "Recoil, — = — — + — =14.00, .071428, .267

we .0735 2.5027

36" Recoil, —

Equivalent throttling area - 4.7 A. A. Trailer, Mgdel

1918
(Calculated)

R = = W* =

175WJ 175Rh

K » — = 1.43, Ka=2.045, A = 9.337 sq.in., A3=913. 994
.7

K«A8 = 1869-U7

863

i t 1" Recoil ••• * 1869-117<16-°60' - .800656, W...447

175x12072.42

A t 4" Recoil* V£ = 1869.117*19.330

176x11330.17

____^*

A t 12" Recoil=W| = 1869'11?Xl6'614 =.315279,WX = .561

175x9350.81

^B^^^^_^B^2

A t 16" Recoil»W| « 1869.117x15.090 ,t290878^w , >531

175x8361.14

A t 20" Recoil=W« = 1869>11?Xl3>245 =. 254184, Wx * .504

175x7371.46 .

1869.117x11.569

A t 24" Recoil=W* = - =. 223998, Wx = .473

175x6381.79

•^^vwft

A t 28" Recoil -W«» 1869.117x9.395 = <174836^w = >418

175x5392.11

9

A t 32" Recoil=W» = 1869' 117x6'604 3 . 105806, «v » .325

175x4402.43

A t 36" Recoil -

Equivalent throttling area - With filling in buffer

Aj, = 1.767 sq.in. AQ * .76 sq.in.

A « 9.337 sq.in., A- = .69 sq.in.

^- = -==»+-=»-^ — +-^—=1.73+2.10=3.83
wb .76 .69 .577 .476

K«(A-Ab)3V«

Wg » -^

„ a -, 1.43* (9.337-1.767)3xl9.33
At 4" Recoil, W| = ,

176(11330 - L33 -1.767 .19.88

Wa

"e

175x.261
2.04x433.76x373.64

1.43 (9.337xl.767)3xl8.03
At 8"Recoil,H| = •

2.04x433.76x325.08
175(10340- ' * 1!

-160' "« • -*00

A« 12"

2.04x433.76x275.89

W = "" *" "• ^^ *^^^m—^m^m^*

1.76.5.51x275.89

-160' "« * _

it 16" Recoil, *. > 1.43'(9.337-1.767)-16.09'

176(8361 . 1.33 .1.

175x.261
a 2.04x433.76x227.7

175(8361 . 1-76>5-"-^-7

45.67

955

r

3_nV

FI6. A

FIG. B

FIG.C

FIG. D

GUN LUGS
PLATE I

966

. 201484_ , _138>
1454775

COH8TBUCTIVI DETAILS.

GUN LUGS - Typical gun lugs are shown in
PLATE I. Plate I, figures A, B, C and 0

respectively. A gun lug properly
speaking is an integral part of a
gun, being an integral part of
the bree ring.

Fig. A, shows an arrangement used on the 75
m/m Futeaux Model 1897. Surrounding the lug is the
piston rod yoke connecting the piston and gun lug,
fig. H - Plate 3. The piston rod yoke and gun lug
are connected by the key passing through both lug
and yoke.

Pig. B, shows a simple construction used on
the 4.7" anti-aircraft mount, model 1917.

Fig. C, shows the lug of the 155 m/ra G.P.F.
gun. The two "holes in the lug are for the
hydraulic and recuperator piston rods.

Fig. D, is a lug for connecting the recoil
sleigh n slide to the gun in the 155 m/m Schneider
Howitzer. The sleigh and gun are further connected
by a front clip, but the pull during the recoil
however, is exerted through the lug, the front
merely supporting the gun.

Two sections are important in the design of

a lug:- the section "ah" just above the rods, which
carries mainly shear, and the section "mn" at the
breech circumference which should be designed main-
ly for bending.

ARRANGEMENT OF GUIDES The recoiling parts
AND CLIPS. are constrained to recoil

ID the direction of the

axis of the bore by the engagement of clips attached
to the gun or recoiling mass, in suitable guides on

967

the cradle or recuperator forging. The reaction
between the guides and clips balance the weight
component of the recoiling parts normal to the
bore and the turning moment, due to the pull of the
various rods about the center of gravity of the
recoiling parts. Due to the large turning moment
caused by the pulls as compared with the weight
component of the recoiling parts normal to the
bore and more or less "play" between the guides and
clips, the normal reactions exerted by the guides
on the clips are more or less concentrated at the
end contacts. The distribution of the bearing
pressure, of course, depends upon the elasticity
and play between the clip and guides, and there-
fore, assumptions based on experience must be made
as to the proper surfaces required. In older
type mounts, we have a continuous clip on the
gun, engaging in the guides of the cradle. Unless
the gun clips are sufficiently long, we have a
varying, (gradually decreasing distance), between
the clip reactions assumed concentrated at the ends
and thus the friction of the guides increases in the
recoil. Due to heating of the guides firing unless
sufficient play is allowed for, warping of the guides
may cause a binding action between the clips and
guides.

Therefore, due to these considerations,

(1) the increase in clip reaction
towards the end of recoil,

and

(2) the difficulty of preventing warp-
ing of the guides or clips

and

(3) the necessity of a long gun
jacket, continuous gun clips have been
nore or less discontinued in modern
artillery.

When gun clips are used we have combinations
of three or more gun clips. When only three

966

clips are used it is possible to maintain practically
only two clips in contact with the guides through-
out the greater part of recoil. This is an advantage
since any warping of the guides, etc., does not
materially effect the operation of the recoil. With
four or more gun clips, we have one or more inter-
mediate clips, thus necessitating a more careful
lining up of the gun clips and design of the guides
to prevent warping unless considerable play is to
be allowed.

Referring to fig. (3) Plate we have an
arrangement of three clips A, 8 and C, which recoil
to an intermediate position A^B^C1, where the
rear clip leaves the guide and the front clip enters
the guide. If the clips are equally spaced as they
should be, this intermediate position is one-half
the length of recoil. In the final position the
clips are in the position A" B" C" at the end of
recoil. If "1" is the distance between clips,
since A should not leave the guide until C enters
the guide and at the end of recoil 6 must be still
in contact with the guide, the length of guide
should be:

Win. length of guides = 2L = b, (3 gun

clips) and therefore,

b

Distance between clips 1 = -

2

With three clips, during the first half of the
recoil, the coordinates with respect to the center
of gravity of the recoiling parts of the front
and rear clips respectively, become those of B and
A while during the latter half, they become those
of C and B.

With four clips we have an intermediate clips
always in contact with the guides; hence a careful

alignment is necessary with more or less to prevent
any binding action of the middle clip and throw
the greater part of the clip load on the extreme

969

front and rear clips respectively . Referring to
fig. (3) Plate ( the clips A,B,C and D

move from the battery position, to the midway
intermediary position, that is when clip A leaves
the guide and clip D just enters the guide. If "1"
is the distance between the extreme clips in bat-
tery, i.e. between A and C, or between B and 0
when clips are equally spaced as they should be,
we have 1 = b, that is the distance between clips
equals the length of recoil. Further the minimum
length of guide = - I - ~ b.

With four clips, the coordinates of the front
and rear clip reactions with respect to the
center of gravity of the recoiling parts during
the first half of recoil become those C and A
respectively, while during the latter half they
become those of 8 and 0 respectively.

Let us now consider the front and rear clip
reactions between the guides and clips of the
gun.

The clip reactions, become, for the front

Clip' P

for the rear clip, Pbs + Bdb*«rcos0(xt-nyx )

where Pb = the max. total powder reaction on the

breech (Ibs)
e = the distance from the axis of the bore

to the center of gravity of the recoil-

ing parts (inches)
B - the total braking pull excluding the

guide friction (Ibs)
dfc = the distance down from the center of

gravity of the recoiling parts to the

line of action of the center of pulls

(inches)

960

Wr = weight of recoiling parts (Ibs)

xt and yt = coordinates of front clip reaction

measured from the center of

gravity of the recoiling parts.
xa and ya = coordinates of rear clip reaction

measured from the center of gravity

of the recoiling parts.
n = coefficient of guide friction = 0.15 approx

0 = the angle of elevation.

1 = xt + x - the distance between clip re-

actions .
Since ny and ny are small as compared with

x and xa respectively, we have for a close approx-

imation,

Pbe+Bdb-Wrcos0.x2 Bdb-Wrcos0 x2

= - • - (approx)

, n

1-2 ndr 1

Pbe+Bdb+Wrcos0 xt Rdb+Krcos

t

(approx)

l-2ndr

where dr a mean distance from center of gravity
of recoiling parts to guide. The
guide friction, becomes,

2nBdb+nWrcos0(x -x. )

RI 3 n(Q +0.) - E — — (Ibs)

l-2ndr

The following table is useful in the layout
arrangement for the gun clips and proper length

of guides, as well as showing the change in
clip reaction and guide friction for ths two
combinations .

961

M

M

X

X

1

1

X

X

^w

M

o

O

o

0

Ll

Ll Ll

t-,

tit

•rt S

•a

c

c c

c

CV3

IN) +

CO

( s q T )

Q

1 -o

1

\ Ml/

'O^

03 TJ

jQ

UOT^OTJjJ 3 p T n £} T * ^ O J.

CD

CD

C

C

•*

O3

X

is

cs

M

X

O

S

O

w

Li

o

f s q T )

tf

o

t,

uot^aeay djig jesg

ja

XXI *

.£>

CD

JQ

N— '

TJ

CVJ

CD

X

9

M

CO

X

0

^Sl

o

8

L!

0

5£

o

( s qt )

1

O Li

£

uoj >o* aa <*T TO »uoj,g

CD

I

ft

03

CO

psatnbsj

.O

sapinS jo q^JJuai *Ufh

-O CO 1 CO

s u o j 3. 3 p 3 j d | i o jeoj pus

^ u o j j u a 3 u ^ o q aouc^efQ

^) 1 03 J3

q i T o o 3 j jo sujta^ u T

sdi-[o ussii^aq souF^sta

XI 1 CM X( 1 CVJ

sdTTO jo »on

CO sf

962

DESIGN AND STRENGTH OF In the design of gun
GUN CLIPS AND GUIDES. clips and guides, the

following points should
be considered: (1) General
considerations as to lay

out, protection from dust, etc; (2) the arrange-
ment of clips and guides as outlined in the previous
paragraph; (3) the computation of the maximum clip
reactions; (4) the design of the clip or guide
for allowable bearing pressure; (5) the strength
of the clip or guides at their various critical
sections, to resist bending, direct stress and
shear.

(1) The location of guides in the
direction normal to the axis of the
bore should be based on the follow*
ing considerations :-

(a) From a cross section of the
gun and recuperator forging,
the best position of guides
and gun clips can be located
with consideration for
minimum stress in gun clips.
This requires that the guides
be located as near the axis

of the bore as possible.

(b) For constructive reasons,
it is good design to keep
the various parts connected
with the recoiling parts as
near the axis of bore as
possible.

(c) The reactions of the guides,
however, are quite independent
of the position of the guides
in a normal direction to the
bore, but since the resisting
section of the cradle or re-
cuperator forging is very large

963

Fk5. F

GUN CLIPS
PLATE 2

964

Q<

t>j

0

-]gu

u

•^Bo

965

as compared with those of the
gun clips; gun clips with
long projections downward from
the gun clip jacket due to
guides too far below the axis
of the bore are undesirable.
Hence the location of the guides depends
upon construction and fabrication features with
due consideration to the strength of the gun clips.
These features in general demand that the guides
be located as close to the axis of the bore as
possible.

(2) For small guns, three clips equally
spaced as described in the previous
paragraph should be used. The front
and rear clips should be bevelled
off, so that smooth entrance may

be made into the guides. Bronze
liners either in the clips or guides
should be used. For larger caliber
guns, more clips should be used
since the clip reactions and cor-
responding friction are reduced.

Considerable tolerance should be
allowed but very careful alignment
made in order to prevent possible
binding .

(3) The computation of clip reactions
has been tabulated in a previous
paragraph, for the common arrange-
ment of either three or four equally
spaced gun clips.

(4) The bearing contact during the
recoil between guides and clips,
depends upon tolerance between the
guides and clips as well as the
elasticity of the material, and on
the magnitude of the wear between

966

the clips and guides. Therefore,

we see the distribution of bearing
pressure and the length of contact
is completely indeterminate. From
practice, however, the following
assumption will be made:

(a) Length of gun clip 1 = 1.8d
(in.) approx. where d = diam.

of bore.

(b) Constant length l'=1.5 d(in.)

(c) Distribution of pressure
assumed triangular.

Therefore, if b1 = contact width of clip and guide,
(inches) we have for the maximum bearing pressure
due to the clip reaction Q (Ibs) .

Q

1 Ibs. per sq.in.
b d

Now the max. allowable bearing pressure steel
on bronze, becomes, pgm = 600 to 800 Ibs. per sq*in.

Hence b' = .0017 to .0022 - (inches)

d

The distance 1-1 ' should be the bevelled length
of clip distributed on either end.

With eccentric pulls the side thrust between
clips and guides causes a bearing reaction Q1 and
if b" is the depth of guides in contact with

clip, we have, b' = .0017 to .0022-=- (inches)

d

(5) The strength of gun clips depends
upon the form or type of gun clip
used. In fig. E, plate 2, we have
the minimum bearing contact (w-x).
The required thickness of the toe
T is based on bending at section
(a-b). Since the front clip re-
action causes this bending, and
the load is divided between two

L.

FIG. H

967

FIQ. K

,
±j

FIG. L.

R5TOM ROD

GUH

Pt-ATE -4-

966

front clips on either side. We have,

, (»-x) /Q (w-x)
1.225 /— 1 = 0.912 / — (in)

where fm = — elastic limit of material used.

STRENGTH OF RECOIL PISTON The greater part
RODS. of recoil piston rods

are subjected to ten-
sion during the re-
coil, and com-
pression during the counter recoil due to the
counter recoil buffer reaction. In a few types
of recoil systems, we have compression in the rod
during the recoil, an example being in the
pneumatic cylinder of the 16" U. S. Railway mount.
The critical diameter of 2 recoil piston rod
is at the smallest section within the gun lug as
shown in figures H, K and L, Plate 4. This diameter
should be based on the recoil pull at maximum
elevation and the inertia load at maximum acceleration
This load is the same that occurs for the gun lug.
Let P a the total fluid reaction + packing friction

on piston and rod (Ibs)
B = total braking (Ibs)

Pb = total max. powder reaction on breech (Ibs)
fm = allowable fibre stress of material used

(Ibs/sq.in)

wp = weight of rod and piston (Ibs)
wr = weight of recoiling parts (Ibs)
d = diameter of smallest free section at gun
lug,

0.7854 f

m

For hollow piston rods, with a "filling in" or
spear buffer chamber, we must consider a section
the greatest distance from the piston but passing
through the buffer for maximum inertia and minimum

969

thickness of the rod. Let w' = weight of piston*

rod to section (Ibs)

dro = outside diam. of buffer rod (in)
drj = inside diam. of buffer chamber (in)
Then using the previous symbols, we have,

dro ~dri = - usually dri is fixed
0.785 fm in consideration of

the buffer design, hence
dro is determined from the above formula.

When piston rods are subjected to compression,
during the counter recoil or with a pneumatic
recuperator during the recoil, the rod should be
treated as a column loaded and constrained at
both ends.

The maximum column load on the rod equals the
maximum counter recoil buffer load, which may be
roughly estimated on the basis of counter recoil
stability at horizontal elevation. If

Cg = constant of counter recoil stability =

0.85 to 0.9

Ws = weight of total gun + carriage (Ibs)
lg = distance from wheel contact to line
of action of Ws, recoiling parts in
battery (in)
b = height of center of gravity of recoil-

ing parts above ground (in)
Bj{ - counter recoil buffer reaction (Ibs)
Fvi = recuperator reaction in battery (Ibs)

R'= approximate total friction (Ibs) = 0.3 W_
W 1 ' W 1 '

then BjJ +R'-Fvi=Cs -2-i from which BJ=Fvi+C '-£-£- - R1

h h

(Ibs)

thus giving the maximum compression load on the
rod.

With pneumatic recuperators if the rod is
under compression, the maximum compression is

970

x,-

«•*- X,

FIG. M

kM-4H

JLL

FKa. M

RECUPERATOR FORGlhSS
Pt-ATE 5

971

liable to be either at the beginning or end of
recoil. At the beginning we have the initial
recuperator reaction + the inertia load of the
rod, and at the end of recoil the maximum recuperator
reaction.

TRUNNIONS AND SUPPORTING In older mounts, the
BRACKETS. trunnions were an integral

part of the gun, the gun
setting directly in the
top carriage. With

mounts using a recoil system between the gun and
top carriage, the trunnions are usually bolted
by a supporting bracket to the cradle, though when
the recuperator becomes a guide support replacing
the necessity of a cradle, the trunnions often are
an integral part of the recuperator forging.

Plate 4 shows recuperator forgings with
trunnions an integral part of the forging, figures
M and P, while fig. N shows a recuperator forging
with a trunnion bracketed on.

Plate 6 shows typical trunnions and their
supporting brackets which are bolted to cradle .

In fig. M, consideration only of the design
of the trunnion itself is necessary, while in
fig. P the strength of section m y should be
considered as well. Section mn is subjected to
bending and shear combined with direct stress.

DESIGN Or TRUNN10KS:

Let w = bearing length of trunnion
Of 3 outside diam. of trunnion
d* * inside diam. of trunnion at section "mn"
f = max. fibre stress, - Ibs. per sq.in.
ft, = allowable bearing pressure - Ibs. per sq.
in.

972

Let w = width of section "ab" just above the rods
w1 = width of section "mn" at the contact of

breech circumference and lug.
dfc, =» the distance down from "mn" to center of

gravity of pulls
d - depth of lugs
T = longitudinal length of lug
Pb - max. total powder pressure on breech
nc = weight of recoiling parts attached to

lug.
wr = total weight of recoiling parts

Then w

B* r<pb-B)

wr
wT = - for section "ob "

r
w'T2 = • for section "ran"

-B)]db
r

6dbfs
If kw = w1, then T = where w = w1 as in

gj £ kf figures A,C and D,

k = 1 and T= -
f

Very often d = 2 db, figures A,C and D, hence

3dfs
T ' ~ where T is given.

fg w

f[B + — (pb-B)l

wr
X and Y * the component reactions of the trunnion

(See Chapter V)

When no rocker is used, the entire trunnion of
width "w" usually has bearing contact in the top
carriage trunnion bearing. The design should be
based on a consideration of both the allowable
bearing pressure f^ and the strength at section

973

"mn" where the trunnion meets the cradle or trun-
nion bracket. We have,

fbw Dt = /X2+Y2 for bearing
J ;. !-• pressure

16 wDt

mn

Combining,

16(X*+Y2)

X2+Ya for strength at section

nf f,

Therefore, assuming dt, we immediately obtain Ot

When, however, a rocker is used the dimensions
depend upon the rocker bearing length. Let

»r = length along trunnion for rocker bear-
ing
wc = length along trunnion for top carriage

bearing
X and Y = top carriage component bearing

reactions

Xr and Yr = component rocker reactions
a= distance from mn to the center of top

carriage bearing
b = distance from mn to the center of rocker

bearing
Mx = the bending moment at section mn in the

plane of the X component reactions.

MV = the bending moment at section mn in the

plane of the Y component reactions.

wc wr

Then w = H-+W- a = w-+ — b = —

2 2

Mx=Xa+Xrb My = Ya + Yrb and M * /H2+M«
then fb«cDt = / X2+Y* at the top carriage bearing
fbwrOt = / XJ+Yf, at the rocker bearing

974

— W

TRUMHIOMS
PLATE 6

976

32 M D*

a,* f , ,_ - Purther , . SSMA^

Since a direct solution for D^ is complicated
a trail solution is preferable. A reasonable
procedure would be to solve for Dt from the bend-
ing equation assuming arbitrarily values for Wy
and Wr. Then knowing Dt approximately we may
solve Wv and Wr in consideration of the allow-
able bearing stress, and then recalculate Dt.

TRUHNIOH BRACKETS:

In the design of trunnion brackets, we have one
of two types:

(1) Where the bracket is secured to

a recuperator forging, the bracket

merely transmitting the trunnion
load to the forging, the Latter of
which is stiff enough to carry the
bending stresses. Plate VI, fig. Q.

(2) Where the bracket is secured to

light built up cradle, as in Plate
VI fig. B.

In the latter case the bracket acts as a
stiffener and takes up the cross wise bending, the
longitudinal shear reaction being transmitted to
the cradle only.

For brackets of type (2), assuming the cradle
merely to take up the shear reaction of the
rivets only, we find a critical section "mn" at the
bottom of the bracket, Fig. R, Plate VI
Section "mn" is subjected to:

(1) Cross-wise bending = Y -

2
d

(2) Longitudinal bending - X -

2

976

(3) A shear stress =

In brackets that are bolted to a recuperator
forging as in fig. Q, Plate VI we have grooves or
guides, which engage in corresponding guides or
grooves in the recuperator forging. The pro-
jections and grooves must be designed to withstand
the allowable total bearing stress and total shear
as well, both of which equal the X component of
the trunnion reaction. The bolts which secure the
brackets to the forging, merely take up the tension,

due the moment caused by the overhang of the trun-
nions and the trunnion reaction.

In the design of trunnion brackets, we have
other critical sections, as ab of fig. Q and cd
of fig. R, that is just above or at the first row
of rivets.

In the design of a trunnion bracket, the critical
section is near the first row of rivets, as sections
ab, cd or mn respectively in fig. R.

The straining action at either one of these
sections consists of:

(a) A direct pressure (or tension)
due to the Y component of the
trunnion reaction.

(b) A shear stress equal to the
X component.

(c) A bending moment in the
longitudinal plane (due to the
moment of the X component = Xr.

(d) A bending moment in a cross
sectional plane due to the Y
component = Y g.

(e) A torsional or twisting
moment due to the X component
* X m.

If now Ix and iy are the moments of inertia of
critical section and dx and dy are the distances to

977

the extreme fibres in the longitudinal and cross-
wise directions, respectively, we have,

xrdx Ygdy Y
f = — : — i — : — ± - for the maximum fibre stress

t — y

To design for the proper distribution and re-

quired strength of rivets, for brackets of type 2,
fig.R, we assume a differential rotation about the
bottom of the bracket. Obviously the shear strain
for the upper rivets is a maximum.

If now, the vertical distance from the bottom
to the top row is r0, for the next lower row r, and
so on, then for the shear in the various rows of
rivets, we have, SQ = c r0, St = c rt, Sa = c ra

etc. Further if we have no rivets for the top

row, n for the next row and so on, then taking
moments about the bottom, we have

c(n0r0+ntr**n8r* --- nnr* ) = X r where r is the
vertical distance from the center line of trunnions

to the bottom of the bracket. Hence

Xr
C = - therefore, assuming nont --- nn

Doro+ntrt " "nnrn respectively and the spacing
of the rows r0rt --- rn respectively, we obtain C.

The shear stress for any one rivet, becomes,

X r r0
S = ——————— for the top rivets,

noro+ntrt~ n rn

for the next row,
--- nnrn

X r r
Sn = ; for the bottom row,

-
The tensions in the rivets or in the bolts as

in fig. Q, are obtained by an exactly similar de-
flection method. - (See design of bolts for pedestal

978

— T*

FK3. S

ELEVATING ARC
PLATE 7

979

FU5. W

FIG X

TOP CARRIAGE

PLATE 8

980

SfCT/OM

C/l/?/?//lG£

PLATE: 9

981

lounts - External Forces).

TOP CARRIAGES. The top carriage sustains the

tipping parts and during the firing
takes up the reactions of tbe
tipping parts. These loads are ap-
plied at the trunnions and elevat-
ing arc respectively. With balancing gear introduced
for high angle firing guns, we have an additional
reaction due to the balancing gear. These loads
are balanced by the supporting forces at the travers-
ing pintle and at some other contact with the base
plate or bottom carriage, the arrangement and
position of which determine to a considerable degree
the type of top carriage used.

With large caliber guns, where the design of
the top carriage depends primarily on strength con-
siderations, special effort should be made to throw
the greater firing load on the trunnions, the ele-
vating arc reaction merely balancing the moment of
the weight of the recoiling parts out of battery.
Then, the elevating gear and balancing gear re-
actions become minor forces as compared with that
sustained at the trunnions. Therefore, in a
preliminary layout, we have the load at the trunnions
balanced by the supporting reactions at tbe pintle
bearing and clip bearing respectively.

Having determined the external reaction, we
may examine critical sections and determine their
respective strengths. By classifying the various
types of top carriages used, certain important sect-
ions and the loadings on then may be pointed oat.

We have tbe following types of top carriages:-

WITH MOBILE ARTILLERY.

(1) Top carriages with side frames
and connected together by transoms
or cross beams supporting the pintle
bearing for traversing the top carriage,
(Plate 8,fig.l)

982

- -|~3-

u

. V

TRAVERSING GEAR

PLATE 10

(2) Pivot yoke type of top carriage,
the pintle bearing fitting in tbe
wheel axle and prevented from over-
turning by a bottom pin fitting in
an equalizer bar tbe latter being
connected to the trail, Pivot yoke
type of top carriage, when used in
a pedestal mount, is prevented fron
overturning by a sufficient shoulder
at tbe top of tbe bearing.

(3) Cantilever top carriage used wben
balancing gear is introduced. Tbe
trunnions being at tbe rear, tbe
pintle bearing at tbe center and front
clip bearing at tbe front, gives a
cantilever loading on tbe top carriage.

WITH TllgP STATIOHABT MOUNTS.

(1) Pivot yoke type of top carriage
with small pedestal mounts.

(2) Side frame top carriages with
barbette mounts.

WITH BAILVA7 ABTILLiRY.

(1) Pivot yoke or side frame top
carriages with pedestal or barbette
mounts, seated on the car frame.

(2) Side frame girders, supporting
tbe trunnions of tbe tipping parts,
directly on tbe girders, and the
girders in turn being supported

by tbe truck reactions, or by a dis-
tributed support of special rail*.
(1) Side Frame Top Carriages:

Side frame top carriages consist of two side
frames either of cast steel or of built up structural
steel. The frames are connected together by a beavy
transom or cross beam which contains tbe pintle bear-
ing for traversing. Tbe pintle bearing and transom

984

are usually located either directly below or to tbe
front of tbe trunnions. Tbe pintle bearing is de-
signed only to take up a part of tbe vertical and
tbe entire horizontal component of tbe reaction of
the tipping parts, tbe overturning moment being
balanced by tbe reactions of either front or rear
circular clips. Either a rack or pinion gear
is introduced at a given radius on tbe top carriage
for traversing about tbe pintle. A pinion or worm
wheel bearing, for tbe pinion or worm engaging in
tbe elevating arc is located at a given radius from
tbe trunnion axis. In tbe design of large guns,
special effort sbould be made to throw tbe greater
load on tbe trunnions, tbe elevating bearing merely
sustaining tbe moment of tbe weight of tbe recoil-
ing parts out of battery.

External Reactions; See Plate H fig.(l)

As a first approximation, assuming tbe entire
firing load to be applied at tbe trunnions, vie have,

2H=K cos 0

p (Ibs) approx.

2V=K sin 0+Wt I

where H and V = the horizontal and vertical load

applied to either trunnion

0.47 wr
K = tbe resistance to recoil = — — — VJ approx

b g

vfv+4700vy
vf ' ~ ; w = weight of shell

w = weight of powder charge

w_ - weight of recoiling parts

~

b = length of recoil (ft)
Wt = weight of tipping parts
Tbe pintle reactions, becomes,
Ha=2H=K cos 0 (Ibs)

2V lt-2H bt lt ht

V_ * -^— — — — = (K sin 0+W*)-r- - K cos0. r—

1

(Ibs)

85

SE:CT\ON A-B
TYPICAL. SECTIONS OF GIRDERS

o o o
o o o

o o o
o o o\

f'.CCCO

PLATE II

986

and the rear clip reaction, becomes,

2Vb«K sin0+Ht-Va (Ibs) for the clip reaction on

either girder.

Structural Steel Sections of Side Frames:

Structural steel side frames are becoming
standardized types of side frames. We have two types
of section, ( 1) box girder types and (2) simple neb
and flange section . The advantage of a box girder
type is that stiffeners are not needed and only one
cross beam or transom is required the frames being
sufficiently rigid. The fabrication of such however
is more complicated, than with simple web and flange
sections.

After a layout contour is made of the frame
several sections should be taken as (m-n) Platell,
fig.U).

A typical box girder shown in Platell, fig. (2)
and a typical flanged web section is shown in Plate
11 fig. (3). As a first approximation it will be
assumed that the flanges take all the bending stress
and the webs the entire shear. If

I = moment of inertia of section

d 3 depth of flange (in)

A = area of flange (sq.in)

fm = max. allowable fibre stress

(Ibs/sq.in)

tbeD d« Ad»

2A — =

- I ap

4 w

d

My Vj

b •'•mn 2

I

v

1

T

b 1ran

A d

1 b mn

- (.sq.in)

(in)

Thus with a constant flange section, we must in-
crease the depth of the girder as the distance from

987

the reaction V^.

If on the other hand for construction considerations
and approximately constant depth of girder is re-
quired, the flange area must be increased with the
distance lmn. These factors determine the cross
section of girder, the area of the flange and the
depth of girder. The depth of girder should "be
sufficient with a given thickness of web "t", not
to exceed the maximum allowable shear stress fs.
Since the shear on the web is practically uniform,
we have, fs dt = V^ for one web and 2fg dt = V^
for two webs as in a box section, then

d = one web, d * • box section

fat 2fgt

Pitch of Rivets in built up girder:

Let p = pitch of rivets (in)

R = allowable total shearing stress on rivat

d = depth of web

F » total shearing force
Then considering a portion of the web of length p

of a compound I section, we have, Fp * Rd, where P=Vb

p j

or the total shear on the section, hence p » — (in)

r

Now for one web and two angle irons connecting the
flange plates with web, we have

n ft

R = 2 - d* fsr * \\ dv fsr where dr =» diaro. of rivet

(in)

fgr * allowable shear
stress on rivet
(Ibs/sq.in)
With a built up box section as in Plate 11 fig. 4,

R - 2 J d« fir - \ d- fsr

fig. (5) R*4- d* fsr = n d£ fsr

4

988

GENERAL DESIGN PROCEDURE.

Type of gun
"Howitzer or Can"

= 155 M/M Howitzer

Type of Mount
"Field Carriage"
"Platform Mount"
"Caterpillar Mount"

= Field Carriage

Diam. of bore
d (inches)

= 6.1

Muzzle Velocity
v (ft/sec)

1850

Weight of Projectile
w (Ibs)

= 95

Weight of ponder charge
w (Its)

14.25

Weight of Recoiling parts

•f

= 4200

Max. pressure on breech
Pbn (Ibs/sq.in)

= 30000

Length of Recoil Max.
Elevation bs (ft)

Max. angle of elevation
0B (degrees)

65'<

Assumed length of
horizontal recoil

(ft)

= 4

S8S

Min. angle of elevation

0i (degrees) --50

Travel of projectile up bore

u (inches) s 117.5

Area of bore

A = 0.786 d* (sq.in) = 29.75

INTERIOR BALLISTICS.

Area of bore of gun
A = 0.785 d'(sq.in)

Total max. pressure on
breech of gun

Mean constant pressure on
projectile

*v'*12 n. ,
Po = -rr: — dfes)

644u

Tine Abscissa of max.
pressure

...c (2^s-i)

16 pe

L- ^- -£=)-U (in)
16 Pe

Pressure on base of breech
when shot leaves muzzle:

(Ibs)

Max. Velocity of free recoil:

wv+4700 ii ..4 , x
Y = (ft/sec)

990

Velocity of free recoil when
shot leaves muzzle:

(w+0.5 w)v
V£o = (ft/sec)

Time of travel of shot to

muzzle:

a. u

Time of free expansion of
gases :

OD

32-2

Free movement of gun while
shot travels to muzzle

<»>

Free movement of gun
during powder expansion

pob fo
xf'o= T~ g "7"+vfotin (ft)

Total free movement of gun
during powder pressure period

E*Xfo+Xf,0 (ft)

Total time of powder pressure

period

T = t »t

991

STABILITY: TOTAL RESISTANCE TO BICOIL AT

MAXIMUM AND MINIMUM ELEVATION,

Weight of system (gun and
carriage) Ws (Ibs)

Distance from spade point
to line of action of YJS
(from preliminary layout)

Height of trunnion from
ground (assume) ht (ft)

Horizontal distance from
spade point to trunnion
center (assume) lt (ft)

Distance from center of
gravity of recoiling
parts to trunnion
(assume) s (ft)

Moment arm of resistance
to recoil for angle of
elevation 0
d = bt cos0+s-ltsin0 (ft)

Height to center of gravity
of recoiling parts for
horizontal recoil
b - h+s (ft)

APPHOIIMATI CALCULATION; (g and T not

comp u t e d)

Velocity of free recoil

wv +47005

Vf -

I WJ

992

Travel up bore u (inches)

Initial recoil constrained
energy (approx)

Ar = - — V* (ft/lbs)

2 8

where Vr = 0.92 Vf (approx)

long recoil
= 0.88 Vf short recoil

Displacement of gun during
powder period

.w+0.5w. u , .
E_ = ( ) — (ft)

wr 12

where a = 2.25 for long recoil
= 2.22 for short recoil

(1) Constant resistance throughout
Recoil .

Constant of horizontal
stability

Overturning moment
Stabilizing moment

(Usually assume 0.85)

Kin. length of recoil con-
sistent with stability at
minimum elevation

U1U p*9V«.b*V/U i

Wsls+WrErcos0-/(Nsls+WrErcos0)2-4Wrcos0(WslsEr-

c

2 YTr cos 0

At 0° Elev. cos 0=1 and d = h

(ft)

Max. alienable recoil at
horizontal elevation

.035 Vf/b (ft)

Assumed length of horizontal
recoil at min. elevation
bb (ft)

Total resistance to recoil at
horizontal or minimum

use Ar for long recoil

Assumed length of recoil at
nax. elevation consistent
Hith clearance bs (ft)

Total resistance to recoil

at nax. elevation (0ffi =

Use Ap for short recoil

993

Variable Resistance to recoil

Constant of horizontal
stability Cs

Min. length of recoil con-
sistent with stability at
min. elevation
1

b

Bf.)] (ft) at 0° elev. cost=l and d=h

994

Max. allowable recoil at
horizontal elevation

bh -.035 Vf /n~ (ft)
"max

Assumed length of recoil
at horizontal or min.
elevation bh (ft)

Mean resistance to recoil
during retardation period

Stability slope
Wrcos0

•JHI

d

Cs • (Ibs/ft)

Mean resistance to recoil
in battery

K -KB+ ~(b-Er) (Ibs)

Mean resistance to recoil
out of battery

k -KB- 5(b-Er) (Ibs)
2

Exact calculation E and T computed (See Interior
Ballistics) .

(1) Constant resistance to recoil.

Constant of stability

(assumed) Cs • =

(C-»0.85 usually)

9

A =W cos 0mi =

996
B*Wrcos«fmin(VfT-E)-Wsls

C * Wsls(VfTHS)+ i -£ V f

cc

Min. length of recoil con-
sistent with stability at
mm. elevation

-B+/B*-4AC /f
— (ft)

Allowable recoil at horizontal
elevation

bh =.035 /h~ (ft)
nmin.

Assumed length of recoil
at minimum elevation
bh (ft)

Total resistance to recoil
at min. elevation

Kh

Max. elevation consistent
with clearance bs (ft)

Total resistance to recoil at
max. elevation

996

(2) Variable resistance to recoil
Constant of stability (assumed) C_ =

WPcos*5
Stability slope ffl=C — (Ibs/ft)

S d

Total resistance to recoil during
powder period consistent with

stability

C8(*8l.-WrE cos 0)

(lbs)

COS

A = m

B , •£!! -2K-2BE

K*mT8

C =(2E 2 VfT)K + — ~-

Br 4ro*

Min. length of recoil
consistent with stability
at min. elevation

• . =Bij£i*L (f t )

Allowable recoil at horizontal
elevation

bh = .035 Vf /~h (ft)
"max-.

997

Assumed length of recoil at
minimum elevation bh (ft)

Total resistance to recoil during
powder period with assumed length
of recoil at min. elevation

mrVf+ai(b-E)2
Kh

2fbh-E+VfT- - — (b-E)]
2 ro

Total resistance to recoil in out
of battery position with assumed
length of recoil at min. elev.

Margin of stability at minimum
elevation for the assumed long
recoil in and out of battery
respectively.

Mean constant pressure on breech
of gun

'be

1.12

Max. overturning force in battery
(stability limit)

l-2ndr

n = 0.15 to 0.25

1 = k , 3 clips = b, 4 clips

Ci

dr = mean distance to guide friction
from bore.

998

Max. overturning force out of
battery

*«l«~Wf bh cos 0
k{ « -2-2 — F. — !]

Margin of stability in battery
K-R (Ibs)

Margin of stability out of
battery k'=k (Ibs)

Estimated Jump of Carriage at Horizontal elevation,

Distance from spade to center

of gravity of Wg — ds (ft) =

Time of recoil (approx.)

"r vf
t « — - — • (sec) =

Ang. vel. about spade at
end of tine

t
g(Knd-Wgls)ti

(rad/sec)

Tine to nax . lift of carriage
from end of time t
d|

t * - w (sec)

Total angular displacement about

spade to max. lift.

i

6 = - w (t +t ) (rad)

999

Lift of wheel from ground
Sw = Iwe=l8 6 (appro*) (ft)
where lw = distance from spade to
wheel base (ft)

Potential energy at nax. lift
E8=WS18 6 (ft/lbs)

VABYIKe TBK RECOIL OB BLKVATIOMt

In general assume the length of recoil at

horizontal recoil constant from £)j to 0} degrees,
(usually from 0° to 20° elevation), then, decrease
the recoil proportionally with the elevation, or
consistent with clearance.

Length of intermediate recoil (ft) from A^ to 0m

b b
b = - (£H0n)+bs

degrees. b b

ri i

bh = long recoil
b8 = short recoil

Resistance to recoil: from 0j to 0j

Variable Kn =

2[bh-E+VfT- - ~(b-E)]

o -

Lj ID ft

]

Constant Kh ** — — — —
b-E+VfT

from £5 2 to ^

assumed constant = K = — -^—— exact

b-E+VfT

1000

0.47

Moment arm of resistance to recoil about
spade d = htcos0+S-ltsin# (ft)

« •»*

•

a •

a.

>

C.'

0 -H
i-l W

0 •

4* M

o *•

V) 41

n 10

I/I

H

M

p> 1i

•H

in i-i

6 o «•

ll +> P -O

4* 0

v w

Q) *rt

(V 0

0

•»i h

h O

• .a

. _

rH •

o

•*»*»*

9 "&

•H 0 6

•H *J •

rH 4)

a i/i «>

ft
00

a

00

a
o

•a

JB -H

<0 *
•»» *
CO >

4> »

(0 C

4» 0
(O O

(0 *•

V

o o

O -rl H «

E «0 •*! »

0 « 0 P.

»-*»••

0.

bh

CsWrcos0i

0

Kh

«i

Usually

J

*

from 0°

•

#

.

•

.

*

to 20°

jj

bh

CsWrCCS01

0

K

Q •

di

01

bh

0

0

Kh

^

From 20°

to max.

•

•

•

•

•

•

elev .

0

b

0

0

K

d

L

i.

0

0

«*

*

1001

Initial recuperator reaction
(appro*. ) (Max. elev. =0m)

Fvi= 1.3 Wf(sin Om+0.3cose(in)(lbs)

5623.8

Min. Bean recuperator reaction
(Max. elev. 0m)

FVB)=2Wrrsin0m-0.3(l-cos0m)+0.3Wrl

(Ibs)

7418

Min. allowable ratio of com-
pression

P^ _ 1.5(1.665Fvm-Fvi)

r,,^ fm*

M

1.79

Max. allowable ratio of com-
pression (stability limit)

"max

VI

VI

= 9.

Max. allowable ratio of com-
pression (heating limit)
m = 2 to 2.5

Mean temperature in recuperator

(assumed)

!„, = 20° to 30° C (centigrade)

Max. temperature due to com-
pression

where k = 1.3 with floating piston
= 1.1 air contact with oil

1002

Ratio of compression used.
M

Max. allowable air pressure
Pafa = 200° to 250°

Final air pressure
paf (Ibs/sq.in)

Initial air oressure

Paf
Pai = " (Ibs/sq.in)

ID

Initial recuperator air

e

P,

volume required:

vi , m .
V_ = bj,(— - — — • ) (cu.in)

Pai i

m

k -

bn * length of horizontal recoil
(inches )

Effective area of recuperator
piston

Fvi

A., = (Ibs/sq.in)

Pai

Length of air column in terns
of recoil stroke

j = 0.8 to 1.2 usually

Actual length of air column
1 * ' j bn (in)

1003

Air cyl. cross section Aa

Ratio of - = —

Effect. area of recuperator Av

Aa 1, m*

r = — = -( — • - ) =2.8

Av j J

• - 1

Area of cross section of air
cylinder

Aa = r Av (sq.in) =

Max. allowable fibre stress in
the recuperator piston rod

f_ (Ibs/sq.in) =

a t
fD = - to - elastic limit usually

Required area of cross section of
recuperator piston rod

= 1.2 -~ (sq.in)

Required diam.of recuperator
piston rod

(in)
0.7864

Assumed diam. of recuperator
piston rod
dv (in)

Area of cross section of re-
cuperator piston rod
a = 0.7864 d

Required area of recuperator

cylinder

A -A +av (sq.in)

1004

Required diao. of recuperator
cylinder

vo

'vo
0.785

(in)

Assumed diam. of recuperator
cylinder dvo (in. )

Area of recuperator cylinder
A (sq.in)

Effective area of recuperator

piston

Av = Avo-av (sq.in)

Initial recuperator pressure

P.

ai

Fvi

= (Ibs/sq.in)

A
v

Final recuperator pressure
Paf = m Pai (Ibs/sq.in)

Initial air volume (exact)
i

''" '

k=1.3 to 1.1

•

Length of air col uran(exact )
1 - - (in)

1006

RECOIL BRAKE LAYOUT.

Max .hydraulic pull (at max. elev.)

-F (Ibs)

Max. hydraulic pull(at 0° elev.)
Phc=Kh-0.3Wr-Fvi (Ibs)

Max. allowable brake pressure

ph max. (Ibs/sq .in)

Ph max = 400° to 500° (lbs/sq. in)

Required effective area of recoil piston

Phm
A = - (sq.in)

Ph max

Reciprocal of contraction factor of
orifice assumed C

Win. recoil throttling area

C A Vr

Bin

max
where Vr = 0.9 Vf (approx.)

Hydraulic brake pressure (at 0° elev.)

pbo
pho = -7- (Ibs/sq.in)

Max. recoil throttling area (at 0° elev.)

C A Vr
Wh max = - where Vr = 0.92 Vf (approx.)

The battery stabilizing moment of counter recoil
WsL,j = 150 to 250 La (inch Ibs.)

1006

L£ * distance from wheel base to Ws

L& - distance from spade to wheel contact

Max. buffer reaction of counter recoil
"SLB

DIMIB8IOR8 07 HOLLOW PISTON BODS.

Max. allowable buffer pressure

pb'm (Ibs/sq.in)

Assume from 1600 to 2500 (Ibs/sq.in)

Area of buffer chamber

A. = JLL (sq.in) «

b

Required inside diam.of piston rod

/ *b
0.7834

Area of inside cross section of rod
Filloux recoil system

Abs 3 wbn (sq.in)

Required inside diam. of piston rod
Filloux recoil system

0.7834

Assumed inside dian. of piston rod
d (in)

1007

Max. allowable fibre stress brake piston
rods fm (Ibs/sq.in1)

^ to J elastic limit (Ibs/sq.in)

Outside diam. of piston rod based on
max. allowable tension

« /df +1.6 — (in)

Outside diam. of piston rod based
on max. hoop compression

° ..

Assumed outside diam. of rod do (in)

Area of total cross section of rod
ar • 0.7854 d« (sq.in) -

(2) Dimensions of Solid piston rods

Max. allowable fibre stress
fB (Ibs/sq.in)

3 1

- to j elastic limit (Ibs/sq.in)

Required area of piston rods

pbm
al « 1.3 •• (sq.in) «

r.

Corresponding diam. rod

1008

Assumed dism. of rod
dr (in)

Area of rod ar = 0.7854 dj
(sq.in)

AREA OF DIAM. OF RECOIL OYL1BDIB

Required area of recoil cylinder
AJ=A'+ar (sq.in)

Corresponding diam. of recoil cylinder

is

Assumed diam. of recoil cylinder
d (in)

Area of recoil cylinder
A. = 0.7864 d* (sq.in)

Effective area of recoil piston
A = Ar-ar (sq.in) =

Max. pressure in recoil cylinder

Phm

max

PRINCIPLE RBACTION8 AND STRB8SEE THROUGH-

MOUNT.

Total resistance to recoil at
max. elevation

imrvr
K, = (Ibs)

1009

Initial recuperator reaction
Fvi=1.3Wr(sin0m+0.3cos0m) (Ibs) =

Max. hydraulic pull (approx.)
max. elev.

FROM RECUPERATOR AND RECOIL BRAKE LAYOUT

DETERKINE:

Distance from axis of bore to line of action of P1

n

dh (in)

Distance from axis of bore to line of action of

dv (in)

Distance from axis of bore to line of action
of resultant braking B

Distance between guide friction 1 (in) =

1 = — for 3 clips (in)
2

1 = bj, for 4 clips (in)

Coordinates along bore of front and rear clip
reactions with respect to center of gravity of

recoiling parts
x± (in)
xt (in)

1010

Distance from axil of bore to line of action
of mean guide friction (from layout) dr (in)

Total braking at max. elevation

Ut-x 1
K.+Wp(sin0-ncosfl *

1

(Ibg)

I
where n * 0.1 to 0.2

Recuperator piston friction

Fvi
Rpv « -04 n Dvo T^- wp

Assume W. • width of packing (in)

Recuperator stuffing box friction

Fvi
R - .04 « dy — - W8 (Ibs)

Ay

Assume wg « width of packing (in)

Total recuperator packing friction
R(e*p)v(lb«>

Hydraulic piston friction

K.

Rph - .04 * D -jp irp (Its)

Hydraulic stuffing box friction
Rib - ,04n dp — w, (Iba)

Total hydraulic packing friction

1011
Total hydraulic pull (max. elev.)

Total hydraulic reaction
(max. elcv.)

Max. hydraulic pressure

phm
phm - (Ibs/sq.in)

A

Max. recuperator reaction

Fvf»m Pvi (Ibs)

where m * ratio of compression

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